Logic puzzle

ABSTRACT

A three-dimensional polyhedra-form logic puzzle comprised of a plurality of independent, non-interconnected puzzle elements of three-dimensional geometric, axially-symmetric, pyramidal-peaked form including one or more apex-opposed polygon faces and a plurality of radial faces. Each edge of an apex-opposed face that contacts a radial face presents visible design indicia. Radial faces contain transversely-polarized magnetic, mechanical or electro-mechanical attachment mechanisms. The puzzle is solved and the three-dimensional polyhedra completed when puzzle elements are placed with the visible design indicia of every puzzle element&#39;s apex-opposed face edges conforming to those of every adjacent puzzle element. Radial faces may contain protuberances and cavities corresponding to indicia of corresponding apex-opposed face edges such that only puzzle elements with conforming apex-opposed face indicia will interjoin. Computerized processes determine viable solutions for indicia combinations on puzzle elements. Puzzle elements may be of pyramidal, dipyramidal, trapezohedral, rhombohedric or other similar forms.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to geometric logic puzzles characterizedby independent, solid, three-dimensional, axially-symmetric,pyramidal-peaked puzzle elements, each of which puzzle elements has oneor more apex-opposed polygonal faces bearing visible indicia. Eachaxially-symmetric, pyramidal-peaked puzzle element may be comprised ofthree-dimensional elements of pyramidal, dipyramidal, trapezohedral,rhombohedric or other similar forms with radial faces of identical shapeand size.

The goal of the puzzle is to assemble a pre-determinedthree-dimensional, solid geometric shape by placing the puzzle elementsin an orientation with the apices of the pyramidal-peaked puzzleelements touching and with the edges of their apex-opposed faces alsotouching. The desired arrangement of the puzzle elements is such thatthe indicia of each of the edges of the visible apex-opposed polygonalfaces of each puzzle element conforms to the indicia of the visibleapex-opposed face edges of all adjacent puzzle elements. The visibleindicia of such puzzles can be alphanumeric values, color elements,abstract patterns or geometric designs. In solving such puzzles, themost challenging ones are those that offer a number of apparentlyviable, nearly-completed solutions using many of the puzzle elements butwhich cannot be fully completed using the remaining puzzle elements.Only a select set of alignments of all of the puzzle elements—preferablyjust a single configuration—yield the completed assembly of thepre-determined geometric shape with full conformance of the visibleindicia on the faces of the geometric puzzle elements.

This invention also relates to geometric logic puzzles that utilizeanalytical systems which may include computerized processes to determinecombinations of indicia on the pyramidal-peaked puzzle elementsresulting in viable, non-trivial, challenging solutions to the assemblyof the pre-determined solid geometric shape. Finally, this inventionrelates to computerized manifestations and representations ofthree-dimensional polyhedra-type geometric logic puzzles with pyramidal,dipyramidal, trapezohedral, rhombohedric or other similar forms ofpuzzle elements that are solved through interaction with computerizedrepresentations of these puzzle elements.

2. Description of the Prior Art

Puzzles have entertained and amused mankind for centuries. The presentinvention relates to geometric puzzles, logic puzzles and puzzles whosepuzzle elements may be interjoined to form a single interconnectedobject.

Geometric puzzles can be as simple as two-dimensional tiles orthree-dimensional building blocks. Alignment of the tiles or blocks cancreate larger geometric units of a predetermined shape.

Logic puzzles are designed and constructed with specifically delineatedgoals and operate within a framework of constraints on the availableoptions. A puzzle solver uses logic to evaluate the typicallyinterrelated constraints in order to ascertain the solution to thespecified goal. A recently popular example of a logic puzzle is known asSudoku. In a Sudoku puzzle, the goal is to fill a 9-by-9 grid with thenumbers 1 through 9 while satisfying three constraints simultaneously.One constraint is that the nine 3-by-3 sub-grids cannot containduplicate entries of the same number. The others are that neither anyrow nor any column through the whole grid can contain a duplicate entryof the same number. An initial Sudoku grid is partially filled withentries and the challenge for the puzzle solver is to logically evaluatethe alternatives and find the single arrangement of missing numbers thatfills all the remaining cells in a manner complying with the puzzleconstraints.

Interconnecting blocks such as LEGO® are an indirect example of a puzzlewhere the puzzle elements may be interjoined to form a singleinterconnected object. The creation of a specific design—such as asailing ship—using LEGO® constitutes a type of interjoining puzzle.

A geometric puzzle may exhibit the logic requirement whilesimultaneously using puzzle elements that may be interjoined. Thesecharacteristics are demonstrated in the well-known jigsaw puzzles, whichhave pictorial representations with repetitive or somewhat ambiguousvisual patterns requiring logic to discern their placement. Such puzzleelements (jigsaw pieces) must satisfy both the visual indicators of thefinal pictorial representation and the physical requirements of thecutout designs necessary for interjoining with adjacent puzzle elements.When all the puzzle elements have been correctly placed, the fullpictorial view is correctly presented and each puzzle element isphysically interlocked with adjacent pieces into a single geometricobject.

Geometric puzzles, logic puzzles and interjoining element puzzles havebeen the subject of numerous patented inventions. More than 125 yearsago, U.S. Design Pat. No. 4,793 (Apr. 11, 1871) was issued to SamuelLoyd (1841-1911) for a logic puzzle that arranged eightdistinctly-shaped, solid, three-dimensional puzzle elements to form acube. Three-dimensional geometric puzzles typically offer theopportunity for the puzzle solver to interact with geometric solidswhere, by nature, only a part of the entire surface of the puzzleelements and/or of the final geometric construct are visible from asingle vantage point. The puzzle solver must either physically ormentally rotate the geometric puzzle elements and/or completed geometricconstruct to evaluate alternative solutions to the puzzle.

One type of three-dimensional geometric puzzles offers the challenge offolding or otherwise manipulating a set of permanently interconnectedcomponents of a solid geometric shape into a configuration thatsatisfies specified conditions. These puzzles are characterized bymultiple pieces that are permanently attached to one another at edges orother exterior points, allowing for folding or flexing the pieces withinthe constraint of the attachment mechanism. U.S. Pat. No. 2,992,829discloses devices that are hinged along their edges and can bemanipulated to create a variety of three-dimensional geometric solids.U.S. Pat. No. 4,323,244 discloses a solid geometrical puzzle comprisedof a plurality of basic components that are hinged along two edges toneighboring components. U.S. Pat. No. 5,338,034 discloses athree-dimensional puzzle that includes multiple, permanently connected,irregular pyramids that are assembled into a regular tetrahedron. Theapices of the irregular pyramids all meet at one point in the interiorof the assembled tetrahedron and the bases of the irregular pyramidsform the regular tetrahedron surfaces. U.S. Pat. No. 5,108,100 disclosesa puzzle comprised of series of three-sided pyramids and eight-sidedoctaeder-shaped bodies interconnected between apex points by a string,the goal of the puzzle being to assemble the interconnected puzzleelements in a manner creating a large tetrahedon with uniformly coloredsides.

A second type of three-dimensional geometric puzzles offers thechallenge of rearranging interconnected but movable components of asolid geometric shape into a configuration that satisfies specifiedconditions. These puzzles are characterized by multiple pieces that arepermanently attached to one another at interior points, allowing forshifting and rotating the pieces within the constraint of the attachmentmechanism. Many of these puzzles provide the solver with the challengeof matching visible characteristics of the exposed surfaces but requiresthat the conformance be accomplished through a sequence shifts androtations of the interconnected puzzle elements. Perhaps the best knownpuzzle of this type is Rubik's Cube®, the goal of which is to arrangethe rotatable puzzle elements so that each cubic face is composed ofpuzzle elements of the same color. U.S. Pat. No. 3,655,201 discloses apattern forming puzzle in the shape of a cube such that interconnectedpuzzle elements on the faces of the cube may be rotated to presentuniformly colored faces on each side of the completed puzzle. Otherexamples of three-dimensional, rotational puzzles are disclosed in U.S.Pat. No. 4,378,116, U.S. Pat. No. 4,473,228, U.S. Pat. No. 4,474,376,U.S. Pat. No. 4,575,088 and U.S. Pat. No. 7,108,263. A variation of therotational puzzle is disclosed in U.S. Pat. No. 4,416,453 in which onlyopposing faces of the polyhedron are interconnected and rotate inunison. U.S. Pat. No. 4,558,866 discloses a three-dimensional,rotational type, polyhedra-form logic puzzle with the solution goal ofmatching visible indicia. Puzzle elements are rotated to interchangetheir positions in the process of arriving at the solution to thepuzzle.

A third type of three-dimensional geometric puzzles offers the challengeof assembling independent three-dimensional puzzle elements into apre-determined solid geometric shape. The challenge of most such puzzlesis finding the arrangements of puzzle elements that create the desiredfinal structure. Loyd's Cube Puzzle is an example of this type ofpuzzle. U.S. Pat. No. 4,676,507 discloses a three-dimensional puzzlecomprised of six identically-shaped puzzle elements that physicallyinterconnect to create one of the five convex regular polyhedragenerally known as the Platonic solids. A Platonic solid is a polyhedronall of whose faces are congruent regular convex polygons and where thesame number of faces meets at every vertex. U.S. Pat. No. 6,145,837discloses a puzzle made of several three-dimensional puzzle elementsthat can be positioned to create several predetermined geometric shapes.U.S. Pat. No. 6,439,571 discloses a puzzle of cubic puzzle elements thathave raised quadrants on some surfaces and flat quadrants on othersurfaces such that the puzzle elements may be placed together to createa line, a square, a cube or a tesseract. U.S. Pat. No. 7,247,075discloses a set of three-dimensional, rhombic-pyramid-shaped buildingblocks. U.S. Pat. App. Pub. No. US2005/0014112 A1 discloses a geometricsolid entertainment system consisting of regular polygon,three-dimensional, pyramid-shaped building blocks.

Because independent three-dimensional puzzle elements offer multiplepossible orientations in space, a fourth type of geometric puzzlescreates solid three-dimensional shapes assembled by fitting together anumber of smaller three-dimensional shapes while satisfying other,constraining conditions on the available alternative orientations of thepuzzle elements. Such constraints allow the puzzle design to include therequirement to demonstrate specific characteristics such as matching thevisible indicia or other physical attributes of adjacent puzzleelements. U.S. Pat. No. 3,565,442 discloses a puzzle composed ofthree-dimensional tetrahedral and octahedral puzzle elements to bearranged such that the visual indicia of die markings on the exteriorfaces are arranged to total the sum of thirteen on each face of thefinal polyhedron. U.S. Pat. No. 3,788,645 discloses a mathematical cubepuzzle in which four separate cubes have on each of their edges one ofthree different colors. The object of the puzzle is to arrange thevarious cubes relative to one another such that the colors associatedwith all exposed adjacent playing edges of different cubes match oneanother. The puzzle has multiple solutions and the pieces can bearranged into a wide variety of different shapes. U.S. Pat. No.4,210,332 discloses a pattern forming puzzle using independent puzzleelements with visible indicia on each face of the puzzle elements thatare placed into a support structure in such a way that a predeterminedpattern is created on each face of the final polyhedron. U.S. Pat. No.4,258,479 discloses a puzzle with sets of right-triangular tetrahedronblocks, such sets being capable of assembly into cubes or pyramidsexhibiting specified color schemes on the exterior faces. U.S. Pat. No.4,865,324 discloses a three-dimensional puzzle comprised of a pluralityof wheel-shaped puzzle elements along a common axis, the puzzle beingsolved when puzzle elements with matching visible indicia are placed inthe correct order and rotated into alignment along that central axis.U.S. Pat. No. 5,407,201 discloses a puzzle with multiplethree-dimensional pieces of tetrahedral and square pyramidal form thatfeature indicia along their edges. When a three-dimensional geometricstructure is correctly assembled from the pieces, completed indiciaappear on all surfaces of the assembled geometric structure, with theportion of the indicia on each piece of the surface matching thecomplementary portion of the indicia on the adjacent surface. U.S. Pat.No. 5,411,262 discloses a puzzle with multiple, essentiallytwo-dimensional puzzle elements that may be assembled to form a hollowthree-dimensional object. U.S. Pat. No. 5,803,461 discloses a game inthe form of a puzzle with two-dimensional or three-dimensional,identically-sized cubic puzzle elements marked by visible indicia. Thegoal of the game/puzzle is to align the puzzle elements such that theindicia match along adjacent edges while forming the geometric shape ofa square. U.S. Pat. No. 6,257,574 discloses a multi-polyhedral puzzle offour tetrahedrons and a single octahedron that are fitted together in atransparent case in a manner such that abutting faces of adjacentpolyhedron blocks form prescribed color patterns. U.S. Pat. No.6,422,560 discloses a puzzle with multiple, three-dimensional cubicpuzzle elements that may be assembled to form a three-dimensional cubewith a distinct composite image on each face of the cube.

All four types of three-dimensional geometric puzzles described aboverequire one or more processes to hold the puzzle elements together inthe finished object. For the interconnected puzzles of the first type,direct physical connection is used to bind the pieces. A direct physicalattachment mechanism is typically used for rotational-type puzzlesalthough some variations, such as magnetic attachment, are suggested asin U.S. Pat. No. 4,558,866. Independent puzzle elements in puzzle typesthree and four are not bound by direct physical connection. Gravity incombination with friction is used in some designs as in U.S. Pat. No.3,565,442. In other cases, the pieces are arranged in a separate supportstructure as in U.S. Pat. No. 4,210,332, U.S. Pat. No. 5,411,262 andU.S. Pat. No. 6,257,574. Magnetic mechanisms are used in a number ofthree-dimensional puzzles, as in U.S. Pat. No. 4,258,479, U.S. Pat. No.5,411,262, U.S. Pat. No. 6,439,571, U.S. Pat. No. 7,247,075 and U.S.Pat. App. Pub. No. US2005/0014112 A1. Velcro, another releasablephysical attachment mechanism, is proposed for securing pieces in placein U.S. Pat. App. Pub. No. US2005/0014112 A1.

Some of the three-dimensional puzzle inventions with independent puzzleelements disclose the use of the physical characteristics of the puzzleelements for assuring that adjacent puzzle elements are correctlyaligned. U.S. Pat. No. 4,865,324 discloses that protuberances andcavities on the magnetic, wheel-shaped puzzle elements hold them inalignment in the direction of the axis so that visible indicia areprecisely positioned to subsequently determine if the indicia arecorrectly matched. U.S. Pat. No. 6,439,571 discloses cubic puzzleelements that have raised quadrants on some surfaces and flat quadrantson other surfaces such that a raised quadrant will align and interlockwith a flat quadrant and the linkage secured by magnetic bondingelements for the purpose of assembling the puzzle elements into thefinal configuration.

None of the inventions described herein, nor any known or discovered,disclose a design for a three-dimensional logic puzzle 1) comprised ofindependent, three-dimensional, axially-symmetric, pyramidal-peakedshaped puzzle elements with clearly identifiable, identically-shapedradial faces that 2) exhibit design indicia on the edges of theirvisibly exposed faces and 3) all attaching radial faces of the puzzleelements contain transversely-polarized, magnetically attractive orother securing components such that 4) every properly aligned attachingradial face is identical in shape to and will be attracted to any otherproperly aligned attaching radial face and 5) that design indicia ofevery visibly exposed edge of every puzzle element must be correctlymatched with its adjoining puzzle elements in order to successfullycomplete the puzzle. Further, none of the inventions described herein,nor any known or discovered, disclose a design for a three-dimensionallogic puzzle that possesses physical characteristics on its attachingfaces corresponding to the design indicia on the exposed faces such thatonly attaching faces with conforming visual indicia on exposed faces maybe physically interjoined to successfully complete the puzzle. Radialfaces of pyramidal-peaked puzzle elements are those faces that share avertex point with the center of the complete polyhedron and an edge withthe exterior of the complete polyhedron. Visibly exposed faces are thosefaces of the puzzle elements that do not share a vertex point with thecenter of the completed polyhedron and can be seen on the exterior ofthe completed polyhedron. The pyramidal-peaked puzzle elements aredesigned such that a clear orientation may be quickly establishedidentifying all potential radial faces and visibly exposed faces forconformance of design indicia.

Further, none of the inventions described herein, nor any known ordiscovered, disclose a process for creating the arrangement of designindicia for a three-dimensional logic puzzle where the design indiciahave been chosen such that all potential solutions to the puzzle for agiven set of puzzle elements are known and can be evaluated to determinedesign indicia that yield a puzzle of desired difficulty, thatdifficulty being indicated by the number of potential solutions to thepuzzle offered by the specific set of puzzle elements or by the numberof partially completed puzzle configurations that cannot be successfullycompleted with the remaining puzzle elements. U.S. Pat. No. 5,407,201discloses a logic puzzle with multiple three-dimensional puzzle elementsof tetrahedral and square pyramidal form that feature indicia alongtheir edges and includes a list of design indicia requirements for somepuzzle element edges that must be met to assure that a solution doesexist for the puzzle. However, because some of those puzzle elements mayhave multiple rotational orientations in all three dimensions, not alldesign indicia are matched with those of other puzzle elements either 1)because the indicia of the puzzle elements are hidden inside thecompleted geometric structure, 2) because the indicia of the puzzleelements are hidden inside the supporting base, or 3) because edgeindicia of puzzle elements along the unbounded edge of the completedgeometric structure are not constrained to match those of other puzzleelements. Thus, no process is disclosed for determining the appropriatedesign indicia for hidden or unbounded puzzle elements and no process isprovided to assure that arbitrary assignment of those non-constraineddesign indicia won't yield a large number of alternative solutions tothe puzzle making successful completion of the puzzle significantly lessdifficult than was expected. Further, no process is disclosed fordetermining the appropriate design indicia for puzzle elements used in apuzzle where every design indicia of every puzzle element visible on theexterior of the completed polyhedron is required to match the designindicia of every adjoining visible puzzle element.

None of the inventions described herein, nor any known or discovered,disclose a design for a three-dimensional logic puzzle with independent,axially-symmetric, pyramidal-peaked puzzle elements wherein every radialface of the pyramidal-peaked puzzle elements is identical in shape toevery other radial face and every properly aligned radial face possessesa transversely-polarized magnetic element that will magnetically attractany other radial face that is similarly aligned. Proper alignment of twopyramidal-peaked puzzle elements is achieved when the two of theidentically-shaped radial faces of the puzzle elements are oriented sothat vertices of the radial faces that both share the center of thecompleted polyhedron and edges of the radial faces that share thevisibly exposed faces are aligned. Every known magnetically-attractive,three-dimensional geometric design using matching faces ofthree-dimensional puzzle elements exhibits the characteristic that agiven face contains a magnet with one of two possible polarities. Thus,for any specific polarity, some puzzle elements will be attracted andothers will not. For instance, in U.S. Pat. No. 7,247,075 a lefttriangular face is designed to attract and attach only to a righttriangular face. U.S. Pat. No. 5,411,262 discloses in FIG. 7 anarrangement of magnets along the edges of essentially two-dimensionalpuzzle elements that causes any edge of any puzzle element to beattracted to any edge of another puzzle element. However, if one of thepuzzle elements is flipped over—an apparently allowable potentialalignment for the two-dimensional puzzle elements—then none of its edgesare attracted to any of the edges of other puzzle elements.

None of the inventions described herein, nor any known or discovered,disclose a design for a three-dimensional logic puzzle with independent,axially-symmetric, pyramidal-peaked puzzle elements wherein every radialface of the pyramidal-peaked puzzle elements—relative to the apex of thepyramid—is readily identifiable and distinguished from its base face andis identical in shape to every other radial face of the other puzzleelements. For instance, in U.S. Pat. No. 7,247,075 some of the radialfaces of the axially-asymmetric pyramidal elements are referred to asleft triangular faces whereas others are described as right triangularfaces. U.S. Pat. No. 5,407,201 discloses a logic puzzle with multiplethree-dimensional puzzle elements of tetrahedral and square pyramidalform, with many of the triangular faces of identical shape. However, thetetrahedral puzzle elements do not present identifiable radial facesthat can be readily distinguished relative to a discernable apex fromthe other faces of the puzzle elements.

None of the inventions described herein, nor any known or discovered,disclose a design for a three-dimensional logic puzzle assuring thatonly attaching faces of puzzle elements with conforming visual indiciamay be physically interjoined. The arrangement of protuberances andcavities on the connecting edges of the puzzle elements disclosed inU.S. Pat. No. 4,865,324 are designed for the sole purpose of assuringthat visible design indicia on the exterior edges of the wheel-shapedpuzzle elements are presented in a limited number of distinct alignmentsrather that offering a nearly unlimited combination of alignmentpossibilities that would result if the puzzle elements turned withoutrestriction. The physical nature of the protuberances and cavities asdisclosed do not, however, prevent puzzle elements with non-matchingvisible indicia from being placed and connected to one another.

None of the inventions described herein, nor any known or discovered,disclose a design for a three-dimensional logic puzzle such that everyedge of every apex-opposed face of every puzzle element must becorrectly matched with the edges of apex-opposed faces of the adjoiningpuzzle elements in order to successfully complete the puzzle. U.S. Pat.No. 5,407,201 discloses a puzzle with independent three-dimensionalpuzzle elements that requires matching visible indicia along the edgesof the puzzle elements. However, many of the edges of puzzle elements inthat puzzle are not constrained to match with the edges of any otherpuzzle elements. These unmatched edges are either hidden inside thecompleted three-dimensional structure (including the supporting base) orsit on the unbounded exterior of the completed structure.

None of the inventions described herein, nor any known or discovered,disclose a design for a three-dimensional logic puzzle such that everypuzzle element is of three-dimensional dipyramidal shape—as though twopyramidal shaped puzzle elements had been securely joined over theirbase faces—with the dipyramidal puzzle elements still offeringrecognizable radial faces to be interjoined based on conformance betweenthe visibly exposed faces of the other halves of the dipyramidal puzzleelements.

None of the inventions described herein, nor any known or discovered,disclose a design for a three-dimensional logic puzzle such that everypuzzle element is of three-dimensional trapezohedral shape—consisting ofidentical deltoid faces—with the trapezohedral puzzle elements offeringrecognizable radial faces to be interjoined based on conformance betweenthe visibly exposed faces of the other halves of the trapezohedralpuzzle elements.

Finally, none of the inventions described herein, nor any known ordiscovered, disclose a design for a three-dimensional logic puzzle suchthat every puzzle element is of three-dimensional rhombohedricshape—consisting of identical parallelogram faces—with the rhombohedricpuzzle elements still offering recognizable apices distinguishing theorientation of the radial faces to be interjoined based on conformancebetween the visibly exposed faces of the other halves of therhombohedric puzzle elements.

SUMMARY OF THE INVENTION

The present invention discloses a geometric logic puzzle characterizedby independent, solid, three-dimensional, axially-symmetric,pyramidal-peaked puzzle elements constructing a completed polyhedra withvisible indicia on the apex-opposed, exposed faces of each puzzleelement and conditions specified for placement of the puzzle elementssuch that visual indicia on adjacent puzzle elements satisfy certainconstraints. Axially-symmetric, pyramidal-peaked puzzle elements—ofpyramidal, dipyramidal, trapezohederal, rhombohedric or other similarform—are those puzzle elements that exhibit a plurality ofidentically-shaped, flat faces angled such that the faces converge at apeak associated with an apex and such that the appearance of the puzzleelement is identical for a face-to-face rotation around the central axispassing through the apex.

A regular pyramid is the pyramidal-peaked polyhedron formed by joiningan apex-opposed regular polygonal base face to the apex with triangularradial faces. An apex-opposed face is a face of a polyhedron that doesnot contact that apex. For axially-symmetric, pyramidal-peaked puzzleelements, every face of the puzzle element either contacts a given apexas a radial face or is opposed to that apex. A dipyramid is thedouble-pyramidal-peaked polyhedron formed by joining a regular pyramidand its mirror image base-to-base making the apex-opposed faces thoseradial faces of the opposite apex. A trapezohedron is thedouble-pyramidal-peaked polyhedron formed by joining uniform deltoids. Arhombohedron is a special case of a trapezohedron for which thedouble-pyramidal-peaked polyhedron is formed by joining uniformparallelograms.

As a puzzle solver selects puzzle pieces to adjoin each other andselects which faces of those pieces to align facing each other in thecompleted puzzle, a face of one puzzle piece can be releasably attachedto an adjacent face of an adjoining puzzle piece by a number of possibleattachment mechanisms. The pyramidal-peaked puzzle elements may containtransversely-polarized magnetic, mechanical or electromechanicalattachment mechanisms on or inside each radial face aligned to securetwo conforming radial faces when they are placed together. If it isdetermined that two puzzle pieces do not properly fit next to each otherin the completed puzzle, the pieces can easily be disengaged from eachother to allow other configurations to be attempted.

Each radial face of each puzzle element can beneficially exhibitphysical characteristics corresponding with the indicium on the edge ofthe apex-opposed face shared by the radial face. Such a radial face maycontain patterns of protuberances and cavities arranged such that itinterjoins with a conforming radial face of another puzzle element—andthereby conforming edges of apex-opposed faces on the two puzzleelements—when the puzzle elements are placed together. The patterns ofprotuberances and cavities prevent interjoining when two puzzle elementswith non-conforming radial faces—and thereby non-conforming edges ofapex-opposed faces—are placed together.

This invention discloses as an example the specific geometric logicpuzzles characterized by independent, solid, three-dimensional,axially-symmetric, pyramidal puzzle elements with visible indicia oneach edge of each identically-shaped base face of each puzzle elementthat are assembled to create a three-dimensional solid polyhedron whilesatisfying specified constraining conditions. Each edge of each regularpolygonal base face of each pyramidal puzzle element expresses visibledesign indicium that can be an alphanumeric value, a color element, anabstract pattern, a geometric design or a combination of thealternatives. By way of example, pyramidal puzzle elements possess asingle distinct apex point and a single, apex-opposed, regular polygonalbase face. Thus, pyramidal puzzle elements may be oriented in a singledistinct direction wherein the sets of triangular radial faces sharevertex points at the apices of the puzzle elements with the center ofthe complete polyhedron and the apex-opposed base faces are visiblyexposed on the exterior of the completed polyhedron. The pyramidalpuzzle elements contain transversely-polarized magnetic, mechanical orelectromechanical attachment units on or inside each triangular radialface aligned to secure two conforming triangular faces when they areplaced together. The three-dimensional polyhedra with conforming indiciaon both sides of each exposed base face edge results when all pyramidalpuzzle elements with conforming triangular radial faces are properlyaligned and interjoined.

This invention also discloses as an example the specific geometric logicpuzzles characterized by independent, solid, three-dimensional,axially-symmetric, dipyramidal puzzle elements with visible indicia oneach identically-shaped triangular face of each puzzle element that areassembled to create a three-dimensional solid polyhedron whilesatisfying specified constraining conditions. Each triangular face ofeach dipyramidal puzzle element expresses visible design indicium thatcan be an alphanumeric value, a color element, an abstract pattern, ageometric design or a combination of the alternatives. Dipyramidalpuzzle elements consisting of only triangular faces are distinguishedfrom pyramidal puzzle elements by possessing two distinct apex pointsinstead of just one and by having no base face. Thus, dipyramidal puzzleelements may be oriented in two distinct directions wherein 1) one setof triangular faces share vertex points at the apices of the puzzleelements with the center of the complete polyhedron and the othertriangular faces are visibly exposed or 2) the apices of the puzzleelements shared by the previously exposed and visible faces are insteadplaced at the center of the complete polyhedron and the previouslyinterior triangular faces are visibly exposed on the exterior of thecompleted polyhedron. The dipyramidal puzzle elements beneficiallycontain transversely-polarized magnetic, mechanical or electromechanicalattachment units on or inside each triangular face aligned to secure twoconforming triangular faces when they are placed together. Thethree-dimensional, stellated polyhedra with conforming indicia on bothsides of each exposed central edge results when all dipyramidal puzzleelements with conforming triangular faces are aligned and interjoined.

A central edge is an edge of the three-dimensional dipyramidal puzzleelement that does not contact either apex. As with the pyramidal puzzleelements, each triangular face can exhibit physical characteristicscorresponding with the indicium on its opposing-edged triangular face.An opposing-edged triangular face is that face of the dipyramidal puzzleelement with which the initial triangular face shares a central edge andwhich face is connected to the opposing apex. The physicalcharacteristics of each triangular face can consist of patterns ofprotuberances and cavities aligned such that the face of a puzzleelement interjoins with the triangular face of another puzzle elementwhen conforming opposing-edged triangular faces of the two puzzleelements are placed together at their central edge. The patterns ofprotuberances and cavities prevent interjoining when an attempt is madeto place two puzzle elements with non-conforming opposing-edgedtriangular faces together.

The completed polyhedra of this invention created from dipyramidalpuzzle elements are referred to as stellated, or star-like, formsalthough they may not be identical to the geometrically-definedstellated form since the dipyramidal puzzle elements need not strictlyexhibit the extension of internal faces or edges. The completedpolyhedra that may be created from dipyramidal puzzle elements includestellated versions of the Platonic solids, the Archimedean solids, andother uniform and non-uniform polyhedra.

This invention also discloses as an example the specific geometric logicpuzzles characterized by independent, solid, three-dimensional,axially-symmetric, trapezohedral puzzle elements with visible indicia onall identically-shaped deltoidal (kite-shaped) faces of each puzzleelement that are assembled to create a three-dimensional solidpolyhedron while satisfying specified constraining conditions. Eachdeltoidal face of each trapezohedral puzzle element expresses visibledesign indicium that can be an alphanumeric value, a color element, anabstract pattern, a geometric design or a combination of alternatives.Trapezohedral puzzle elements are distinguished from pyramidal anddipyramidal puzzle elements by possessing only deltoidal faces but likedipyramidal puzzle elements they possess two distinct apex points and nobase face. Thus, trapezohedral puzzle elements may be oriented in twodistinct directions wherein 1) one set of deltoidal faces share vertexpoints at the apices of the puzzle elements with the center of thecomplete polyhedron and the other deltoidal faces are visibly exposed or2) the apices of the puzzle elements shared by the previously exposedand visible faces are instead placed at the center of the completepolyhedron and the previously interior deltoidal faces are visiblyexposed on the exterior of the completed polyhedron. The trapezohedralpuzzle elements contain transversely-polarized magnetic, mechanical orelectromechanical attachment units on or inside each deltoidal facealigned to secure two conforming deltoidal faces when they are placedtogether. The three-dimensional polyhedra with conforming indicia onboth sides of each exposed central edge results when all trapezohedralpuzzle elements with conforming deltoidal faces are aligned andinterjoined. The completed polyhedra that may be created fromtrapezohedral puzzle elements include alternately-stellated versions ofthe Platonic solids, the Archimedean solids, and other uniform andnon-uniform polyhedra

A central edge is an edge of the three-dimensional trapezohedral puzzleelement that does not contact either apex. As with the pyramidal anddipyramidal puzzle elements, each deltoidal face can exhibit physicalcharacteristics corresponding with the indicium on its twoopposing-edged deltoidal faces. Opposing-edged deltoidal faces are thosetwo faces of the trapezohedral puzzle element with which the initialdeltoidal face shares a central edge and which faces are connected tothe opposing apex. The physical characteristics of each deltoidal facecan consist of patterns of protuberances and cavities aligned such thatthe face of a puzzle element interjoins with the deltoidal face ofanother puzzle element when conforming opposing-edged deltoidal faces ofthe two puzzle elements are placed together along their central edges.The patterns of protuberances and cavities prevent interjoining when anattempt is made to place two puzzle elements with non-conformingopposing-edged deltoidal faces together.

This invention also discloses as an example the specific geometric logicpuzzles characterized by independent, solid, three-dimensional,axially-symmetric, rhombohedric puzzle elements with visible indicia onall identically-shaped parallelogram faces of each puzzle element thatare assembled to create a three-dimensional solid polyhedron whilesatisfying specified constraining conditions. Each parallelogram face ofeach rhombohedric puzzle element expresses visible design indicium thatcan be an alphanumeric value, a color element, an abstract pattern, ageometric design or a combination of alternatives. Rhombohedric puzzleelements are distinguished from pyramidal and dipyramidal puzzleelements by possessing only parallelogram faces but like dipyramidalpuzzle elements they possess two distinct apex points and no base face.Thus, rhombohedric puzzle elements may be oriented in two distinctdirections wherein 1) one set of parallelogram faces share vertex pointsat the apices of the puzzle elements with the center of the completepolyhedron and the other parallelogram faces are visibly exposed or 2)the apices of the puzzle elements shared by the previously exposed andvisible faces are instead placed at the center of the completepolyhedron and the previously interior parallelogram faces are visiblyexposed on the exterior of the completed polyhedron. The rhombohedricpuzzle elements ideally contain transversely-polarized magnetic,mechanical or electromechanical attachment units on or inside eachparallelogram face aligned to secure two conforming parallelogram faceswhen they are placed together. The three-dimensional polyhedra withconforming indicia on both sides of each exposed central edge resultswhen all rhombohedric puzzle elements with conforming parallelogramfaces are properly aligned and interjoined. The completed polyhedra thatmay be created from rhombohedric puzzle elements include the rhombichexecontahedron.

A central edge is an edge of the three-dimensional rhombohedric puzzleelement that does not contact either apex. As with the pyramidal anddipyramidal puzzle elements, each parallelogram face can exhibitphysical characteristics corresponding with the indicium on its twoopposing-edged parallelogram faces. Opposing-edged parallelogram facesare those two faces of the rhombohedric puzzle element with which theinitial parallelogram face shares a central edge and which faces areconnected to the opposing apex. The physical characteristics of eachparallelogram face can consist of patterns of protuberances and cavitiesaligned such that the face of a puzzle element interjoins with theparallelogram face of another puzzle element when conformingopposing-edged parallelogram faces of the two puzzle elements are placedtogether along their central edges. The patterns of protuberances andcavities prevent interjoining when an attempt is made to place twopuzzle elements with non-conforming opposing-edged parallelogram facestogether.

This invention discloses the use of transversely-polarized magnetic,mechanical or electro-mechanical attachment units on or inside theattaching faces of each three-dimensional, axially-symmetric,pyramidal-peaked puzzle element so that any puzzle element that iscorrectly aligned with an adjoining puzzle element is secured tocomplete the final polyhedra and yields a three-dimensional object thatcan be rotated, examined and handled without the constraint of asupporting base or structure.

This invention discloses the use of physical characteristics on theattaching faces of each puzzle element consisting of patterns ofprotuberances and cavities such that only properly aligned puzzleelements with conforming visible indicia on the exposed faces may besuccessfully interjoined. Those patterns of protuberances and cavitiesprevent interjoining when an attempt is made to place the properlyaligned faces of two non-conforming puzzle elements together.

The development of an analytical solution determination process thatevaluates the possible configurations of indicia on a set of puzzleelements is a significant component of the invention. Someconfigurations of indicia yield multiple solutions while otherconfigurations yield no solutions. For example, if every edge of everypuzzle element were to contain the same indicium, any puzzle elementcould be interjoined with any other puzzle element in any rotationalalignment, making the solution of the puzzle extremely trivial.Conversely, if every edge of every puzzle element were to contain adistinct indicium, then no two puzzle elements could ever be combined,making no solution possible whatsoever. The more challengingconfigurations of indicia on the puzzle elements 1) would contain thesame indicium on the edges of many of the puzzle elements, 2) would usea combination of indicia on the exposed face of each puzzle element thatwas distinct from the combination on any other puzzle element, and 3)would choose the set of puzzle elements such that a limited arrangementof aligned puzzle elements yielded the correct solution to the puzzle.For the most difficult configuration, successful placement of eachpuzzle element would be interdependent on the correct placement of everyother puzzle element, yielding just a single solution to completion ofthe puzzle.

This invention discloses a plurality of three-dimensional polyhedra thatare used for the completed puzzle. Many of the final polyhedra resultingas solutions for the puzzles of this invention contain so many faces andedges that the choice of indicia configurations for the puzzle elementsis a difficult task. This invention discloses an analytical solutiondetermination process that can be effectively implemented bycomputerized techniques to determine combinations of indicia on anygiven set of puzzle elements that result in viable, non-trivialsolutions to the assembly of the pre-determined solid polyhedral shape.

In addition, an evaluation process is disclosed to evaluate the numberof decision steps required in determining the arrangement of puzzleelements yielding a viable solution to the puzzle. In solving thepuzzle, a number of conforming alignments that offer apparently viablepartial solutions may be found that employ some—but not all—of thepuzzle elements. Many of those partial solutions cannot be brought tocompletion using the remaining puzzle elements. A final solution canonly be found from a select set of alignments of all of the puzzleelements. The greater the number of partial solutions offered by theconfiguration of indicia on the puzzle elements, the greater the numberof decision steps (on average) that are required to determine a solutionto the puzzle. The measure of the number of potential decision stepsprovides a strong indicator of the challenge involved in solving thepuzzle.

This invention discloses the use of visual indicia on the pyramidal,dipyramidal, trapezohedral, rhombohedric or other similar puzzleelements. In some implementations, the visual indicia may be acompounded form of two or more types of indicia (such as abstractpatterns placed in a background of color elements) with each type ofindicia chosen in such a way as to represent a distinct puzzle. A singleset of such puzzle elements can be used to provide two or moredistinctly different puzzle solving challenges with distinctly differentlevels of difficulty.

Finally, this invention discloses that computerized manifestations andrepresentations of the three-dimensional, polyhedra-type, geometriclogic puzzles of this invention with pyramidal, dipyramidal,trapezohedral, rhombohedric or other similar puzzle elements can be usedto solve the puzzles of this invention through human interaction withcomputerized representations of the puzzle elements.

OBJECTIVES AND ADVANTAGES OF THE INVENTION

The present invention discloses a logic puzzle of three-dimensionalpolyhedra-form consisting of independent, three-dimensional,axially-symmetric, pyramidal-peaked puzzle elements with the puzzleelements of the required dimensions and present in the exact numbernecessary and sufficient to construct a predetermined three-dimensionalpolyhedron. The objective of the puzzle is to position the puzzleelements relative to each other in a way that satisfies specifiedconditions on the relationships of adjacent visual indicia on thevisible portions of adjoining puzzle elements. For every particularmanifestation of this puzzle, all connecting radial faces of all puzzleelements for a specified polyhedron have exactly the same dimensionsmaking every radial face identical in shape and size. Thus, each puzzleelement presents an identical physical shape over a series of rotationsaround the central axis of the pyramidal-peaked puzzle element (i.e.,axially-symmetric), the number of identical appearances equal to thenumber of radial faces of the puzzle element. Puzzle elements can be ofpyramidal, dipyramidal, trapezohedral, rhombohedric or other similarform with identical triangular, deltoidal, parallelogrammatic or otherradial faces, respectively. Each radial face of each puzzle elementbeneficially contains transversely-polarized magnetic, mechanical orelectromechanical attachment components on or inside that face so thatany puzzle element that is correctly aligned with an adjoining puzzleelement is secured to complete the final polyhedra. The radial faces ofthe puzzle elements can also exhibit physical characteristics consistingof patterns of protuberances and cavities such that only properlyaligned puzzle elements with conforming visible indicia on the exposedfaces may be successfully interjoined and those puzzle elements withnon-conforming visual indicia may not be successfully interjoined.

The objective of the disclosed invention is to provide an interesting,challenging, enjoyable, educational, entertaining, and visuallyappealing puzzle that can be solved by a wide range of puzzleenthusiasts by offering a range of complexity from relatively easy tovery difficult. This invention also offers tactile and visualinteraction with the three-dimensional puzzle elements and completedobject.

An advantage of these requirements is that a very large number ofcombinations of placements of a set of puzzle elements are possible inconstructing the finished polyhedron. Using twelve 5-pyramid puzzleelements (having a regular pentagon as the base face of the pyramidalpuzzle element) to construct a twelve-sided dodecahedron and startingwith any one of the puzzle elements, there are 1,949,062,500,000,000(about 2 quadrillion) possible positional and rotational arrangements ofthe eleven remaining puzzle elements. This result is the product of five(5) to the eleventh (11^(th)) power times eleven factorial (11!). Thelarge number of combinations makes this puzzle potentially verychallenging but the constraining conditions on the conforming of visualindicia on the puzzle elements limits the number of those combinationsthat successfully solve the puzzle. The puzzle solver must use logicaldeduction and inference regarding the constraining conditions and theavailable puzzle elements to effectively solve the puzzle because theprocess of random selection and placement of puzzle elements would beessentially futile in finding a successful solution to the puzzle. Theprocess of discovering a single solution from the vast number ofpossibilities gives the puzzle solver a significant sense ofaccomplishment.

An alternative implementation of this invention using twenty 3-pyramidpuzzle elements to construct a twenty-sided icosahedron has 1.4×10²⁶(about 140 septillion) possible positional and rotational arrangementsof the nineteen puzzle elements that remain after arbitrarily choosingone puzzle element as the starting point.

A further object of the invention is to provide axially-symmetric,pyramidal-peaked puzzle elements containing attachment mechanisms, suchas transversely-polarized magnets, so that appropriately aligned puzzleelements may be interjoined over their conforming radial faces and thepartially completed polyhedron may be rotated, examined and evaluated todetermine the subsequent choices of puzzle elements for placement inattempting to complete the puzzle. The attachment mechanismsbeneficially offer the advantage that the completed polyhedron isself-supporting and needs no framework or other structural components tomaintain its final shape. Ideal attachment mechanisms allow for easyrelease of puzzle elements which have been placed adjacent to eachother, to allow for new placement of puzzle elements if an attemptedarrangement is not successful.

A further possible object of the invention is to provideaxially-symmetric, pyramidal-peaked puzzle elements which exhibitphysical characteristics on the radial faces such as patterns ofprotuberances and cavities corresponding to the visual indicia on thebase face edges to which the radial faces are attached such that onlytwo puzzle elements with conforming visual indicia on the edges of theirexposed faces may be successfully interjoined on their correspondingradial faces, providing a physical indication and confirmation of thesuccessful conforming of the visual indicia of the puzzle elements.

A further object of this invention is to provide pyramidal, dipyramidal,trapezohedral, rhombohedric or other similar form puzzle elements whichmay be used to create convex polyhedra such as the dodecahedron,stellated polyhedra with triangular faces such as the stellateddodecahedron, alternately-stellated polyhedra with deltoidal faces,stellated polyhedra with parallelogrammatic faces, and other variouspolyhedra that present completed puzzles with distinct and appealingappearances.

This invention discloses a plurality of three-dimensional polyhedra thatcan be used for the completed object of the puzzle; many of thesepolyhedra contain many faces and edges making the selection of indiciaconfigurations for the puzzle elements a difficult task. An advantage ofthis invention is that an analytical solution determination process isdisclosed that can be used to determine the patterns of visual indiciaon a set of puzzle elements, assuring that the constraints specified bythe puzzle conditions (e.g., conforming edges) can be met and that asolution to the puzzle is possible. Another advantage is that adisclosed evaluation process can be used to estimate the difficulty incompleting the puzzle for any given set of puzzle elements, permittingthe design of puzzles of various levels of difficulty.

Another advantage of this invention is that the visual indicia of thepyramidal, dipyramidal, trapezohedral, rhombohedric or other similarform puzzle elements may be compounded to allow for multiple expressionsof alphanumeric values, color elements, abstract patterns or geometricdesigns in such a way that each type of expression of indicia representsa distinct puzzle. Thus, a single set of puzzle elements can be used toprovide two or more distinctly different puzzle solving challenges withdistinctly different levels of difficulty.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a front-side view of a completed dodecahedron with edgespresenting conforming indicia of shading patterns.

FIG. 2 shows a set of detailed views of a completed dodecahedron withnumbered faces and with edges presenting conforming indicia of shadingpatterns. FIG. 2 a is the front view, FIG. 2 b is the back view, FIG. 2c is the top view and FIG. 2 d is the bottom view.

FIG. 3 shows a set of detailed views of a 5-pyramidal puzzle element.FIG. 3 a shows the base face with lower case alphabetic-labeled edgesprogressing around the face in a clockwise direction. FIG. 3 b shows aside view of the 5-pyramidal puzzle element with arrows showing thelocations of lower case alphabetic-labeled edges on the obscured bottombase face.

FIG. 4 shows the base face of a 5-pyramid puzzle element with theshading pattern indicia labeled with upper case letters.

FIG. 5 shows a flat view of a dodecahedron with the numbered facesshowing lower case alphabetic-labeled edges and the sequence of facesprogressing around the first face outward in a clockwise direction.

FIG. 6 shows a flat view of a completed dodecahedron with the numberedfaces showing upper case alphabetic-labeled indicia combinations thatprovide a solution to a puzzle for that set of indicia combinations.

FIG. 7 shows a flat view of a completed dodecahedron with the numberedfaces showing upper case alphabetic-labeled indicia combinations thatprovide a solution to a puzzle for a different set of indiciacombinations.

FIG. 8 shows a flat view of a partially completed dodecahedron with thenumbered faces showing upper case alphabetic-labeled indiciacombinations that provide a partial solution to a puzzle with adifferent arrangement of the same indicia combinations as in FIG. 7.

FIG. 9 shows a table format of the numbered faces showing lower casealphabetic-labeled adjacent edges that are located between facescorresponding to arrangement of FIG. 5.

FIG. 10 shows a table format of the numbered faces showing upper casealphabetic-labeled indicia combinations that provide a solution to apuzzle for the set of indicia combinations of FIG. 6.

FIG. 11 shows a table format of the numbered faces showing upper casealphabetic-labeled indicia combinations that provide a solution to apuzzle for the set of indicia combinations of FIG. 7.

FIG. 12 shows a table format of the numbered faces showing upper casealphabetic-labeled indicia combinations that provide a partial solutionto a puzzle for the set of indicia combinations in the arrangement ofFIG. 8.

FIG. 13 shows a set of detailed views of transversely-polarized magneticattachment mechanisms on the radial faces of a 5-pyramidal puzzleelement. FIG. 13 a shows the view looking down on the puzzle elementfrom above the apex. FIG. 13 b shows a side view of the 5-pyramidalpuzzle element.

FIG. 14 shows a flat view of the arrangement of transversely-polarizedattachment mechanisms on the radial faces of six 5-pyramidal puzzleelements.

FIG. 15 shows a set of detailed views of protuberances and cavities on aradial face of a pyramidal puzzle element. FIG. 15 a shows a view of theradial face looking down on the puzzle element from above radial face.FIG. 15 b shows a view of the radial face looking at the puzzle elementfrom the base face end of the radial face. FIG. 15 c shows a view of theradial face looking at the puzzle element from the side of the radialface. FIG. 15 d shows a pair of conforming faces with widely-spaced,matching protuberances and cavities viewed from the base face end of theradial face. FIG. 15 e shows a pair of conforming faces withwidely-spaced, matching protuberances and cavities viewed from the sideof the radial face. FIG. 15 f shows a pair of conforming faces withcentrally-spaced, matching protuberances and cavities viewed from thebase face end of the radial face. FIG. 15 g shows a pair of conformingfaces with narrowly-spaced, matching protuberances and cavities viewedfrom the base face end of the radial face.

FIG. 16 shows a side view of the protuberances and cavities inconjunction with transversely-polarized attachment mechanisms on theradial faces of a 5-pyramidal puzzle element.

FIG. 17 shows a set of detailed views of a 3-pyramidal puzzle element.FIG. 17 a shows the base face with alphabetic-labeled edges progressingaround the face in a clockwise direction. FIG. 17 b shows a side view ofthe 3-pyramidal puzzle element with arrows showing the locations ofalphabetic-labeled edges on the obscured bottom base face.

FIG. 18 shows a flat view of an icosahedron with the numbered facesshowing lower case alphabetic-labeled edges and progressing around thefirst face outward in a clockwise direction.

FIG. 19 shows a set of detailed views of a 4-pyramidal puzzle element.FIG. 19 a shows the base face with alphabetic-labeled edges progressingaround the face in a clockwise direction. FIG. 19 b shows a side view ofthe 4-pyramidal puzzle element.

FIG. 20 shows a set of detailed views of a 6-pyramidal puzzle element.FIG. 20 a shows the base face with alphabetic-labeled edges progressingaround the face in a clockwise direction. FIG. 20 b shows a side view ofthe 6-pyramidal puzzle element.

FIG. 21 shows a set of detailed views of an 8-pyramidal puzzle element.FIG. 21 a shows the base face with alphabetic-labeled edges progressingaround the face in a clockwise direction.

FIG. 21 b shows a side view of the 8-pyramidal puzzle element.

FIG. 22 shows a set of detailed views of a 10-pyramidal puzzle element.FIG. 22 a shows the base face with alphabetic-labeled edges progressingaround the face in a clockwise direction.

FIG. 22 b shows a side view of the 10-pyramidal puzzle element.

FIG. 23 shows a front-side view of a completed stellated dodecahedronconstructed using 5-dipyramidal puzzle elements with exposed radialfaces presenting conforming indicia of shading patterns.

FIG. 24 shows a set of detailed views of a 5-dipyramidal puzzle element.FIG. 24 a shows a transparent side view of the 5-dipyramidal puzzleelement. FIG. 24 b shows a side view of the 5-dipyramidal puzzle elementwith alphabetic-labeled radial faces progressing around the puzzleelement in a clockwise direction when viewed from the apices.

FIG. 25 shows a side view of the transversely-polarized magneticattachment mechanisms on the radial faces of a 5-dipyramidal puzzleelement.

FIG. 26 shows a side view of the protuberances and cavities on theradial faces of a 5-dipyramidal puzzle element.

FIG. 27 shows a side view of the protuberances and cavities inconjunction with transversely-polarized attachment mechanisms on theradial faces of a 5-dipyramidal puzzle element.

FIG. 28 shows a set of detailed views of a 3-dipyramidal puzzle element.FIG. 28 a shows a transparent side view of the 3-dipyramidal puzzleelement. FIG. 28 b shows a side view of the 3-dipyramidal puzzle elementwith alphabetic-labeled radial faces progressing around the puzzleelement in a clockwise direction when viewed from the apices.

FIG. 29 shows a transparent side view of a 4-dipyramidal puzzle element.

FIG. 30 shows a transparent side view of a 6-dipyramidal puzzle element.

FIG. 31 shows a transparent side view of an 8-dipyramidal puzzleelement.

FIG. 32 shows a transparent side view of a 10-dipyramidal puzzleelement.

FIG. 33 shows a front-side view of a completed alternately stellateddodecahedron constructed using 5-trapezohedral puzzle elements with eachhalf of every exposed deltoid face presenting shading pattern indiciaconforming to that of the deltoid face of the adjacent puzzle element.

FIG. 34 shows a set of detailed views of a 5-trapezohedral puzzleelement. FIG. 34 a shows a transparent side view of the 5-trapezohedralpuzzle element. FIG. 34 b shows a side view of the 5-trapezohedralpuzzle element with alphabetic-labeled radial face indicia pairsprogressing around the puzzle element in a clockwise direction whenviewed from the apices.

FIG. 35 shows a flat view of a stellated dodecahedron with the numberedradial face indicia pairs showing lower case alphabetic-labeled edges.

FIG. 36 shows a side view of the transversely-polarized attachmentmechanisms on the radial faces of a 5-trapezohedral puzzle element.

FIG. 37 shows a side view of the protuberances and cavities on theradial faces of a 5-trapezohedral puzzle element.

FIG. 38 shows a side view of the protuberances and cavities inconjunction with transversely-polarized attachment mechanisms on theradial faces of a 5-trapezohedral puzzle element.

FIG. 39 shows a front-side view of a completed rhombic hexecontahedronwith each half of every exposed parallelogram face presenting shadingpattern indicia conforming to that of the parallelogram face of theadjacent puzzle element.

FIG. 40 shows a set of detailed views of a rhombohedric puzzle element.FIG. 40 a shows a shaded, horizontal side view of the rhombohedricpuzzle element. FIG. 40 b shows a quasi-transparent view of therhombohedric puzzle element from above an apex of the puzzle element.FIG. 40 c shows a quasi-transparent, vertical side view of therhombohedric puzzle element.

FIG. 41 shows a set of detailed views of a rhombohedric puzzle elementwith each half of every exposed parallelogram face presenting shadingpattern indicia. FIG. 41 a shows a view of two of the exposedparallelogram faces of the rhombohedric puzzle element presentingshading pattern indicia as seen from above an apex of the puzzleelement. FIG. 41 b shows a vertical side view of three of the exposedparallelogram faces of the rhombohedric puzzle element presentingshading pattern indicia. FIG. 41 c shows a view of an exposedparallelogram faces of the rhombohedric puzzle element presentingshading pattern indicia as seen from below the alternate apex of thepuzzle element.

FIG. 42 shows a set of detailed views of a rhombohedric puzzle elementwith each half of every exposed parallelogram face presentingalphabetic-labeled indicia. FIG. 42 a shows a view of the parallelogramfaces of the rhombohedric puzzle element presenting alphabetic-labeledindicia as seen from above an apex of the puzzle element. FIG. 42 bshows a vertical side view of three of the exposed parallelogram facesof the rhombohedric puzzle element presenting alphabetic-labeledindicia. FIG. 42 c shows a view of the parallelogram faces of therhombohedric puzzle element presenting alphabetic-labeled indicia asseen from below the alternate apex of the puzzle element.

FIG. 43 shows a flat view of an icosahedron-like rhombic hexecontahedronpuzzle with each numbered puzzle element showing one set ofalphabetic-labeled edges and progressing around the first puzzle elementoutward in a clockwise direction.

FIG. 44 shows a side view of the protuberances and cavities on theparallelogram faces of a rhombohedric puzzle element.

FIG. 45 shows a side view of the protuberances and cavities inconjunction with transversely-polarized magnetic attachment mechanismson the parallelogram faces a rhombohedric puzzle element.

FIG. 46 shows a flat view of an octahedron.

FIG. 47 shows a flat view of a cuboctahedron.

FIG. 48 shows a flat view of a small rhombicuboctahedron.

FIG. 49 shows a flat view of an icosidodecahedron.

FIG. 50 shows a flat view of a rhombicosidodecahedron.

FIG. 51 shows a flat view of a truncated icosahedron.

FIG. 52 shows a flat view of a great rhombicuboctahedron.

FIG. 53 shows a flat view of a rhombitruncated icosidodecahedron.

FIG. 54 shows a detailed view of the base face of a 5-pyramidal puzzleelement with two distinct sets of indicia.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

The present invention discloses a three-dimensional polyhedra-form logicpuzzle consisting of a plurality of independent, non-interconnectedpuzzle elements that are assembled to create a three-dimensional solidpolyhedron while satisfying specified constraining conditions. Eachpuzzle element is a solid, three-dimensional geometric,axially-symmetric, pyramidal-peaked form that includes one or moreapex-opposed polygon faces and a plurality of identical-shaped radialfaces, each of which puzzle elements has one or more apex-opposedpolygonal faces bearing visible indicia. Each axially-symmetric,pyramidal-peaked puzzle element may be comprised of three-dimensionalelements of pyramidal, dipyramidal, trapezohedral, rhombohedric or othersimilar forms with radial faces of identical shape and size. Much of thediscussion herein will refer to the three-dimensional puzzle elements aspyramidal, having a polyhedron for a base, triangular radial faces and asingle, common vertex shared by all of the radial faces in anaxially-symmetric manner. Alternative examples of axially-symmetric,pyramidal-peaked puzzle elements using dipyramidal, trapezohedral andrhombohedric forms will also be provided. It is understood that theparticular shape and size of each puzzle element, each apex-opposed faceand each radial face may vary, as long as the puzzle elements in aparticular puzzle have radial faces that are of like shape and size.Thus, each reference to a particular shaped element, apex-opposed faceor radial face is intended to be an example and not limiting. As aspecific example, a single, square, apex-opposed face would yield anaxially-symmetric 4-pyramid with a square base—the commonly recognizedEgyptian pyramid form—while a single, regular, pentagonal base facewould yield an axially-symmetric 5-pyramid.

Each edge of each apex-opposed face of each puzzle element expresses atleast one visible design indicium that can be an alphanumeric value, acolor element, an abstract pattern, a geometric design or somecombination of these alternatives. The pyramidal-peaked puzzle elementsbeneficially contain transversely-polarized magnetic, mechanical orelectro-mechanical attachment units on or inside each radial facealigned to secure two matching radial faces when they are placedtogether.

The three-dimensional polyhedra with matching indicia on both sides ofeach exposed visible edge results when all pyramidal-peaked puzzleelements with matching radial faces are aligned and interjoined.Pyramidal-peaked, axially-symmetric puzzle elements can be of pyramidal,dipyramidal, trapezohedral, rhombohedric or other similar forms. Thecompleted polyhedra that can be created from a specifically-proportionedset of pyramidal-peaked puzzle elements include—but are not limitedto—the Platonic solids (such as the dodecahedron and the icosahedron);the Archimedean solids (such as the cuboctahedron and therhombicosidodecahedron); other uniform polyhedra such as the rhombichexecontahedron and the ditrigonal icosidodecahedron; non-uniformpolyhedra such as the elongated square gyrobicupola; a number of convexrhombic polyhedra such as the rhombic dodecahedron; and stellated andalternately-stellated versions of many of these polyhedra. Examples ofpolyhedra that may be formed upon completion of a puzzle are shown inFIGS. 46-53.

A variety of constraining conditions which must be satisfied to completethe puzzle are possible. In the preferred embodiment, the solution tothe puzzle is the assembly of a pre-determined three-dimensional, solidpolyhedron by placing pyramidal-peaked, axially-symmetric puzzleelements of regular pyramidal form in an orientation with the apices ofthe pyramids touching and with an alignment that satisfies the conditionthat the visual indicia on each edge of the visible, apex-opposed,polygonal base face of each puzzle element conforms to the visualindicia on the visible base face edges of all adjacent puzzle elements.One solid polyhedron of the preferred embodiment is the twelve-sideddodecahedron, a front view of which is shown in FIG. 1 with the shadingpattern indicia shown to be matching across each edge. FIG. 2 showsseveral views of the same dodecahedron again with matching patternindicia and with the twelve sides (faces) numbered in a clockwise mannerfrom the top down for reference. FIG. 2 a shows the front view with face1 on top. FIG. 2 b is the back view while FIG. 2 c is the top view andFIG. 2 d is the bottom view. Face 2 is shown on the left in FIG. 2 a andproceeding clockwise around the back of the dodecahedron are shown faces3 and 4 in FIG. 2 b with faces 5, 6 and 7 shown in FIG. 2 a; faces 8, 9and 10 shown in FIG. 2 b; face 11 shown in FIG. 2 a; and face 12 shownin FIG. 2 d. The clockwise orientation of the numbering scheme is shownfrom the top in FIG. 2 c and the counter-clockwise numbering whenlooking up from the bottom (making the sequence opposite in directionfrom the top-down scheme) is shown in FIG. 2 d.

In the preferred embodiment yielding a dodecahedron, the base face ofeach pyramidal puzzle element 10 is a regular pentagon 11 as shown inFIG. 3 a and the resulting 5-pyramid is an axially-symmetric rightpyramid where a line of height h 14 joining the centroid of thepentagonal base face 12 and the apex of each pyramid 13 is perpendicularto that base face as shown in FIG. 3 b. Using a set of twelveappropriately-proportioned 5-pyramidal puzzle elements expressingspecifically-chosen indicia, the completed dodecahedron shown in FIG. 1can be constructed with matching visual indicia on each side of thethirty exposed visible edges of the dodecahedron. For a dodecahedron ofthe preferred embodiment, the appropriate proportions for the5-pyramidal puzzle elements is that the height h 14 of the puzzleelement is a factor of 1.113516 times the length of a side s 15 of thepentagonal base face.

The visual indicia of the invention can consist of alphanumeric values,color elements, abstract patterns or geometric designs. In a preferredembodiment, each base face edge of each 5-pyramidal puzzle elementexhibits one of five distinct color/pattern elements as shown, forexample, in FIG. 4 using the shading pattern indicia shown on thecentral base face of FIG. 1 (also labeled as face 6 in FIG. 2 a). Theshading pattern indicia is shown with an inset alphabetic character (incaps) for reference to denote each particular and distinctive shadingpattern as follows: dark shading A 16, crosshatch shading B 17, whitedots on dark shading C 18, black dots on light shading D 19 and lightshading E 20. This represents one sequence of five distinct shadingpattern indicia. Each of the twelve 5-pyramidal puzzle elements in FIG.1 and FIG. 2 presents a distinct sequence of shading pattern indiciacombinations in a clockwise direction when viewing the base facedirectly.

The puzzle elements are assembled by placing the puzzle elements in anorientation with the apices of each pyramid touching and with analignment such that the visual indicia on the edges of the pentagonalbase face of each puzzle element conform to the visual indicia on thebase face edges of all adjacent puzzle elements as shown in FIG. 1 andFIG. 2. The conformance requirement for the preferred embodiment is thatthe indicia on adjacent edges match each other but other conformancerequirements—such as having numbered indicia additively total a specificvalue—are possible. The conformance requirement that the edges mustsatisfy can be visually verified by the puzzle solver.

FIG. 5 shows a flat view of the arrangement of the twelve pentagonalfaces of a dodecahedron of the preferred embodiment. The edges of eachnumbered face are labeled with alphabetic characters from a through efor reference use in the analytical solution and evaluation processes.The alphabetic numbering proceeds in a clockwise manner around the facewith the a-labeled edge adjacent to the lowest numbered face surroundingthat face. So, for example, the a edge of face 1 is adjacent to face 2and the a edge of face 2 is adjacent to face 1. Similarly, the a edge offace 7 is adjacent to face 2 and the a edge of face 12 is adjacent toface 7. If—as in the preferred embodiment—each face exhibits a uniquesequence of indicia, then a solution to the puzzle will have thecharacteristic that adjacent faces exhibit identical indicia. Forexample, edge 1 a and edge 2 a will exhibit the same indicia as will 1 band 3 a, 2 c and 7 a, 7 d and 12 a, etc., until all thirty edge pairsare identically matched. The property of matching edge-pairs is used indetermining solutions to the puzzle and in evaluating the difficulty infinding any given solution.

This invention discloses an analytical process to determine combinationsof indicia on the puzzle elements resulting in viable, non-trivial,challenging solutions to the assembly of the pre-determined solidgeometric shape. Some potential configurations of indicia can yieldmultiple solutions while other potential configurations can yield nosolutions whatsoever. For example, if the chosen indicia for the puzzleelements were coloration and if every edge of every puzzle element wereto contain the same indicium—blue, for instance—then any puzzle elementcould be placed adjacent to any other puzzle element in any rotationalalignment, making the solution of the puzzle extremely trivial.Conversely, if the chosen indicia for the puzzle elements were numbersand if every edge of every puzzle element were to contain a distinctnumber, then no puzzle element could ever be placed adjacent to another,making no solution possible whatsoever.

For the preferred embodiment of 5-pyramidal puzzle elements completingthe dodecahedron, five distinct indicia are chosen—one for each of thefive edges of each puzzle element. Without repeating any indicium on anypuzzle element and eliminating rotationally-identical alternatives,there are twenty-four possible combination of five distinct indiciarepresented alphabetically as follows: ABCDE, ABCED, ABDCE, ABDEC,ABECD, ABEDC, ACBDE, ACBED, ACDBE, ACDEB, ACEBD, ACEDB, ADBCE, ADBEC,ADCBE, ADCEB, ADEBC, ADECB, AEBCD, AEBDC, AECBD, AECDB, AEDBC, AEDCB.From these twenty-four possible combinations of indicia, twelve must beselected that provide a viable solution to the puzzle.

An example of a set of twelve combinations of indicia that yield asolution to the puzzle of the preferred embodiment is as follows: ABCDE,ABCED, ABDCE, ABDEC, ABECD, ABEDC, ACBDE, ACBED, ACDBE, ACEBD, ADCBE,and AECBD. These indicia combinations complete a conforming dodecahedronin the following arrangement as shown in FIG. 6: ABCDE, ACDBE, ABECD,ABDEC, ACBDE, ACEBD, ABCED, ACBED, AECBD, ADCBE, ABDCE, and ABEDC.

Another example of a set of twelve combinations of indicia that yield asolution to the puzzle of the preferred embodiment is as follows: ABCDE,ABDEC, ABECD, ACBDE, ACDEB, ACEBD, ADEBC, ADECB, AEBCD, AEBDC, AECBD,and AECDB. These indicia complete a conforming dodecahedron in thefollowing sequence as shown in FIG. 7: ABCDE, ADEBC, AEBCD, ACEBD,ABECD, ABDEC, AEBDC, AECDB, ACDEB, ACBDE, AECBD, and ADECB.

This set of combinations of indicia has an alternative conformingarrangement using eight combinations of indicia as shown in FIG. 8:ABCDE, ABDEC, ADEBC, ACEBD, AEBCD, ABECD, AECBD and AEBDC. However, thefour remaining combinations—ACBDE, AECDB, ACDEB and ADECB—cannotcomplete the remainder of this puzzle. Specifically, the combination ofindicia necessary for face 9 requires the partial clockwise sequenceBDC. None of the remaining combinations contains that partial sequence.Similarly, the combination of indicia necessary for face 10 requires thepartial clockwise sequence ED. None of the remaining elements containsthat partial sequence either. Combinations necessary for faces 11 and 12require the partial clockwise sequences EDB and DC, respectively. Again,none of the remaining elements contains those partial sequences. Thus,FIG. 8 shows an arrangement of eight of the available combination ofindicia that partially completes the puzzle but further progress incompleting the puzzle with the remaining indicia combinations is notpossible.

The flat view of the dodecahedron shown in FIG. 5 demonstrates whichedges of each face are adjacent to those of other faces. The edgerelationships of FIG. 5 can be completely represented in a table asshown in FIG. 9. For each face listed on the column at the left, theedges (indicated as the face number followed by the edge letter in lowercase) that are adjacent to the faces shown across the top are shown inthe corresponding cells of the table. For example, edge 1 a of face 1 isadjacent to face 2. Similarly, edge 1 b is adjacent to face 3 and edge 2e of face 2 is adjacent to face 3. The table has the added advantagethat reading down a column labeled across the top gives the edges thatare adjacent to that face. For example, the column labeled 1 shows thatedges 2 a, 3 a, 4 a, 5 a and 6 a are adjacent to face 1. The processembodied in this table discloses a method for find combinations ofindicia that provide a solution to the puzzle of the preferredembodiment.

The example solution of FIG. 6 is shown in the table in FIG. 10. Theedge indicators in lower case letters are shown in the upper left ofeach cell while the representation of the indicia are shown in uppercase letters in the lower right of each cell. Thus, the indicia for face1 of FIG. 6 (ABCDE) are shown across the first row of the table in FIG.10. To satisfy the matching constraint of the preferred embodiment, theedges adjacent to face 1 (reading down the first column) must have thesame sequence of indicia (ABCDE) as the first row. Reading across thesecond row (face 2), the a edge must exhibit the indicium A (adjacent toface 1). Looking across the row, edge e of face 2 (adjacent to face 3)cannot exhibit the indicium B since the indicium for the preferredembodiment are not repeated on any given face. Similarly, edge b cannotexhibit indicium E. Twenty-three possible combinations of indicia remainafter eliminating the combination used for face 1. Of thosetwenty-three, only thirteen satisfy the constraints on face 2; thecombination ACDBE (reading clockwise around the face, i.e., in abcdeorder) chosen for face 2 in the example of FIG. 6 is one of thosethirteen. That same sequence for the edges adjacent to face 2 is placedin the second column. For the third row, face 3 must exhibit B and Eindicia on edges a and b, respectively, and edge e (adjacent to face 4)cannot exhibit indicia C (since face 4 already exhibits C on the edgeadjacent to face 1). Thus, only four possible combinations remain aspossibilities for face three—BEACD, BEDCA, BECAD and BECDA (or theirrotation equivalents of ACDBE, ABEDC, ADBEC and ABECD). The combinationof indicia chosen for face 3 in the example of FIG. 6 is one of thosefour (ABEDC). Similarly restrictions of the choices of the combinationsof indicia for the remaining faces can be obtained from the table inFIG. 10 until a set of choices is found that satisfies all theconstraining conditions applicable to a particular puzzle. In thisexample of the preferred embodiment, the chosen conditions are that nospecific combination is exhibited on more than one face, each faceexhibits each of the five capital-letter-identified indicia only once,and all thirty edges exhibit the same indicia on each of the adjacentfaces.

There are a very large number of possible arrangements of distinctcombinations of indicia. For the twelve faces of the dodecahedron withfive non-repeating indicia on each face, there are over two millionarrangements of twelve distinct indicia combinations. While manuallyfinding the solution to indicia configuration choices is possible usingthe process described and exhibited in FIG. 10, the manual process is adifficult task. However, the solution process of this invention can beimplemented by computerized techniques to effectively and rapidlydetermine arrangements of indicia on any given set of puzzle elementsthat yield viable solutions. From all the arrangements of twelvecombinations of five distinct indicia of the preferred embodiment, onlythree hundred and seventy-two sets (out of over two millionpossibilities) yield viable solutions to the puzzle. Fifty-two of thosehave multiple distinct arrangements of puzzle elements that solve thepuzzle, so only three-hundred and twenty offer a single arrangement ofpuzzle elements as a solution where successful placement of each puzzleelement is interdependent on the correct placement of every other puzzleelement.

In solving the puzzle of the preferred embodiment and other puzzleconfigurations disclosed in this invention, the most challenging onesare those that offer the largest number of apparently viable partialsolutions using some—but not all—of the puzzle elements. Another exampleof a complete solution was shown in FIG. 7. The solution table for thatexample is shown in FIG. 11. However, even starting with the samecombination of indicia for the first face (ABCDE), the arrangement shownin FIG. 8 demonstrates a partial solution to the puzzle using eightpuzzle elements that cannot be completed with the remaining puzzleelements. The partial solution table for the example of FIG. 8 is shownin FIG. 12, which facilitates the examination of the remaining puzzleelements and rejects their success in completing the puzzle. For everyset of twelve combinations of indicia that provide a distinct solutionto the puzzle, a number of arrangements of those combinations yield onlypartial solutions to the puzzle. Completion of a table as shown in FIG.12 for all possible arrangements of the combinations of indicia allows adetermination the number of faces that can be matched with the availablecombinations of indicia (such as the eight faces that were successfullyarranged in the example of FIG. 12) before an impasse is reached. Acrossall sets of indicia combinations, the greater the average number offaces that can be arranged before an impasse is reached the moredifficult the puzzle comprised of that set of indicia is to complete.

The evaluation process of this invention examines each set ofcombinations of indicia that have been determined to provide a singlesolution to the puzzle. One indicia combination of the set is chosen asthe starting point and all possible sequence arrangements of theremaining combinations are evaluated to determine the average number offaces that can successfully matched before an impasse is reached. Thosesets of combinations of indicia with lower average numbers are concludedto provide puzzles with easier solutions while those with higher averagenumbers are concluded to provide puzzles that are more difficult tosolve. Because exhaustive examinations of multiple alternatives aretedious and time-consuming to perform by hand, the evaluation process ofthis invention can be implemented most efficiently by computerizedmeans.

Ideally, the constructed polyhedra of this invention do not utilize asupporting structure or framework. Therefore, the puzzle elements of thecurrent invention beneficially employ magnetic, mechanical orelectromechanical processes to secure matching puzzle elements. In thepreferred embodiment, the 5-pyramidal puzzle elements containtransversely-polarized magnetic attachment units on or inside eachradial face placed to secure two correctly aligned radial faces whenthey are placed together. In the absence of the protuberances andcavities described below, each radial face of any 5-pyramidal puzzleelement could be magnetically secured to any radial face of any other5-pyramidal puzzle element. This is in stark contrast to every otherknown puzzle design using magnetic mechanisms to securethree-dimensional puzzle elements since every other design uses eithersingly polarized magnetic elements or “flippable” double polarizedmagnetic elements such that some aligned puzzle elements are attractedto only certain other puzzle elements and will not attract and secureany and all other puzzle elements. In the preferred embodiment, thetransversely-polarized magnetic attachment units can be applied inconjunction with the arrangement of protuberances and cavities to assurethat only correctly matched puzzle elements are securely interjoined incompleting the polyhedron.

The placement of the transversely-polarized magnetic attachment units 21on a 5-pyramidal puzzle element 10 in the preferred embodiment is shownin FIG. 13. The top view looking down onto the puzzle element from abovethe apex is shown in FIG. 13 a while the side view is shown in FIG. 13b. Note that the arrangement of the polarization (shown as N for northand S for south) is identical for each radial face of the puzzleelement. Were another puzzle element to be placed adjacent to the oneshown with the radial faces aligned (base face edges touching and apicestogether), the N from one would attract and attach with the S from theother while the S would attract and attach with the N. Such would be thecase for any and all aligned puzzle elements as shown in a flat view inFIG. 14. Through this attachment, the completed dodecahedron could beassembled.

Because of the three-dimensional nature of the completed polyhedron, itmay be difficult for all the constraints to be simultaneously observedand verified visually. Accordingly, in the preferred embodiment, thepuzzle elements can exhibit physical characteristics to assure that onlymatching puzzle elements may be aligned and interjoined. These physicalcharacteristics take the form of an arrangement of protuberances andcavities on the radial faces of the 5-pyramidal puzzle elements suchthat the protuberances will securely fit into matching cavities and suchthat the arrangement of protuberances and cavities on each radial facecorrespond to the visual indicium on the base face edge shared by thatradial face. Each cavity serves as a snugly fitting receptacle for aprotuberance. Thus, only the radial faces of 5-pyramidal puzzle elementswith matching base face edges may be successfully interjoined. Radialfaces with non-matching base face edge indicia may have an arrangementof protuberances and cavities that prevent interjoining.

FIG. 15 provides a detailed look at a set of protuberances and cavitieson a radial face of a puzzle element. A view from above the radial faceis shown in FIG. 15 a with the widely-spaced protuberance 22 shown onthe left of the radial face and the corresponding cavity 23 on theright. The same widely-spaced protuberance and cavity set is shown fromthe base face end of the radial face in FIG. 15 b and from the side ofthe radial face in FIG. 15 c. A matching pair of widely-spacedprotuberance and cavity sets is shown in FIG. 15 d where it is clearthat the protuberance on the upper radial face will fit into the cavityon the lower face and the protuberance on the lower radial face will fitinto the cavity on the upper one. A side view of that same pair is shownin FIG. 15 e. A centrally-spaced pair of protuberance and cavity sets isshown in FIG. 15 f while a narrowly-spaced pair is shown in FIG. 15 g.

Three radial faces of a 5-pyramidal puzzle element 10 withtransversely-polarized magnetic attachment units 21 and protuberance andcavity sets 24 on each face are shown in FIG. 16. Other polyhedra inaddition to the dodecahedron may be used as the puzzle objective of thisinvention. For example, another Platonic solid—the icosahedron—may beconstructed from twenty 3-pyramidal puzzle elements 25 of the type shownin FIG. 17 where the height of the pyramid h 26 is equal to 0.755761times the length s 27 of an edge of the triangular base face. It shouldbe noted that while each 3-pyramid puzzle element has four triangularfaces, the 3-pyramid puzzle element used to construct an icosahedron isnot a regular tetrahedron with all sides equal since the radial edges ofthat 3-pyramid puzzle element are about 5% shorter than the edges of thetriangular base face. FIG. 18 shows a flat view of an icosahedron using3-pyramidal puzzle elements with the numbered faces showing lower casealphabetic-labeled edges. The 3-pyramidal puzzle element can also beused to construct the octahedron, another Platonic solid. Thearrangement shown in FIG. 18 for the icosahedron corresponds to that ofFIG. 5 for the dodecahedron and allows for the solution and evaluationprocesses of this invention to be used in a similar manner to discoversolutions for and to evaluate the difficulty of an icosahedron-typepuzzle.

While some polyhedra such as the dodecahedron and the icosahedron useidentical puzzle elements, several polyhedra may be constructed from acombination of appropriately-proportioned pyramidal puzzle elements with3, 4, 5, 6, 8 or 10 base face edges. A pyramidal puzzle element of fourbase sides 28—a 4-pyramid—is shown in FIG. 19 where the ratio of theheight of the pyramid h 29 is a factor of the length s 30 of an edge ofthe square base face specified for the characteristics of the polyhedronfor which that puzzle element will be used. Similarly, FIG. 20 shows6-pyramidal puzzle element 31, FIG. 21 shows an 8-pyramidal puzzleelement 32 and FIG. 22 shows a 10-pyramidal puzzle element 33. Thecharacteristic ratio relating the height and edge length of thesepyramidal puzzle elements is specific to the completed polyhedron thatwill be constructed with those puzzle elements.

The current invention includes geometric logic puzzles characterized byindependent, solid, three-dimensional dipyramidal puzzle elements withvisible indicia on each identically-shaped triangular face of eachpuzzle element that are assembled to create a three-dimensional solidpolyhedron while satisfying specified constraining conditions. One suchpolyhedron—the stellated dodecahedron—is shown in FIG. 23.

Each apex-opposed triangular face of each dipyramidal puzzle elementexpresses a visible design indicium that can be alphanumeric values,color elements, abstract patterns or geometric designs. Dipyramidalpuzzle elements consisting of only triangular faces are distinguishedfrom pyramidal puzzle elements by possessing two distinct apex pointsinstead of just one and no base face. A dipyramidal puzzle element 34with five central edges—a 5-dipyramid—is shown in FIG. 24 forconstruction of a stellated dodecahedron. The characteristic ratio ofthe height h 35 from the center of the puzzle element to either apexrelative to the edge length s 36, as shown in detail in FIG. 24 a, isidentical to the characteristic ratio of 1.113516 from the 5-pyramidused to construct a regular dodecahedron. The edges along the center ofthe dipyramidal puzzle element 34 and the associated apex-opposed facesare labeled with lower case alphabetic characters from a through e andfrom f through j to designated visual indicia as shown in FIG. 24 b forreference use in the analytical solution and evaluation processes. Thealphabetic numbering proceeds in a clockwise manner around the puzzleelement when looking down from above the exposed apex with the opposedapex placed in the center of the completed polyhedron. Were the puzzleelement to be flipped, i.e., rotated such that the positions of theapices were traded, then the edges f through j would be used in place ofthe a through e exposed edges in the analytical solution and evaluationprocesses.

As is shown in FIG. 24, a dipyramidal puzzle element 34 may be orientedin two distinct directions wherein 1) one set of triangular faces sharevertex points at the apices of the puzzle elements with the center ofthe complete polyhedron and the other, apex-opposed, triangular facesare visibly exposed or 2) the apices of the puzzle elements shared bythe previously exposed and visible faces are instead placed at thecenter of the complete polyhedron and the previously interior triangularfaces are visibly exposed as apex-opposed faces on the exterior of thecompleted polyhedron.

Each dipyramidal puzzle element 34 may beneficially containtransversely-polarized magnetic attachment units 37 on or inside eachtriangular face as shown in FIG. 25. As with the pyramidal puzzleelements, each triangular face of a dipyramidal puzzle element 34 canexhibit physical characteristics corresponding with the indicium on itsco-edged triangular face. A co-edged triangular face is that face of adipyramidal puzzle element 34 with which the initial triangular faceshares a central edge and which face is connected to the opposing apex.A central edge is an edge of the three-dimensional dipyramidal puzzleelement 34 that does not contact either apex. An example of the physicalcharacteristics of each triangular face that can be exhibited aspatterns of protuberances and cavities 38 is shown in FIG. 26. Theprotuberances and cavities are aligned such that the face of adipyramidal puzzle element 34 interjoins with the triangular face ofanother dipyramidal puzzle element when matching co-edged triangularfaces of the two dipyramidal puzzle elements are placed together attheir central edge. The patterns of protuberances and cavities preventinterjoining when an attempt is made to place two dipyramidal puzzleelements with non-matching co-edged triangular faces together. Adipyramidal puzzle element 34 with both the transversely-polarizedmagnetic attachment units 37 and several patterns of protuberances andcavities 38 on the triangular faces is shown in FIG. 27.

Three-dimensional polyhedra with matching indicia on both sides of eachexposed central edge results when all dipyramidal puzzle elements withmatching triangular faces are aligned and interjoined. The completedpolyhedra from dipyramidal puzzle elements are referred to in thisinvention as stellated, or star-like, forms although they are notnecessarily the result of geometric stellation since the dipyramidalpuzzle elements need not strictly exhibit the extension of internalfaces or edges of the polyhedra.

The solution and evaluation processes of this invention are used in thesame manner to discover solutions for and evaluate the difficulty of astellated dodecahedron puzzle constructed from dipyramidal puzzleelements in exactly the same manner as they were used for the standarddodecahedron of the preferred embodiment.

Other stellated polyhedra such as the stellated icosahedron may becreated from 3-dipyramidal puzzle elements 39 shown in FIG. 28 where thecharacteristic ratio of the height h 40 from the center of the puzzleelement to either apex relative to the edge length s 41, as shown indetail in FIG. 28 a, is identical to the characteristic ratio of0.755761 from the 3-pyramid used to construct a regular icosahedron. Theedges of each end of the dipyramidal puzzle element are labeled withlower case alphabetic characters from a through c and from d through fas shown in FIG. 28 b for reference use in the analytical solution andevaluation processes. The alphabetic numbering proceeds in a clockwisemanner around the puzzle element when looking down from above the apexof each end. The 3-dipyramidal puzzle element can also be used toconstruct the stellated octahedron, another Platonic solid.

Other dipyramidal puzzle elements such as the 4-dipyramidal puzzleelement (shown in FIG. 29), the 6-dipyramidal puzzle element (shown inFIG. 30), the 8-dipyramidal puzzle element (shown in FIG. 31) and the10-dipyramidal puzzle element (shown in FIG. 32) can be used toconstruct other stellated polyhedra. Those polyhedra include stellatedversions of the Archimedean solids and other uniform and non-uniformpolyhedra.

The current invention also includes geometric logic puzzlescharacterized by independent, solid, three-dimensional trapezohedralpuzzle elements with visible indicia on each identically-shaped deltoidface of each puzzle element that are assembled to create athree-dimensional solid polyhedron while satisfying specifiedconstraining conditions. One such polyhedron—an alternately-stellateddodecahedron—is shown in FIG. 33.

Each half of each apex-opposed deltoid face of each trapezohedral puzzleelement expresses visible design indicium that can be alphanumericvalues, color elements, abstract patterns or geometric designs.Trapezohedral puzzle elements consisting of only deltoid faces aredistinguished from pyramidal puzzle elements and are similar todipyramidal puzzle elements by possessing two distinct apex pointsinstead of just one and no base face. A trapezohedral puzzle element 42with ten central edges—a 5-trapezohedron—is shown in FIG. 34 forconstruction of an alternately-stellated dodecahedron. Thecharacteristic height h from the center of the puzzle element to eitherapex is identical to that of the 5-dipyramidal puzzle element used toconstruct a stellated dodecahedron. The edges along the center of thetrapezohedral puzzle element and associated apex-opposed faces arelabeled with lower case alphabetic characters from a through j and fromk through t to designate visual indicia as shown in FIG. 34 b forreference use in the analytical solution and evaluation processes. Thealphabetic numbering proceeds in a clockwise manner around the puzzleelement when looking down from above the exposed apex with the opposedapex placed in the center of the completed polyhedron. Were the puzzleelement to be flipped, i.e., rotated such that the positions of theapices were traded, then the edges k through t would be used in place ofthe a through j edges in the analytical solution and evaluationprocesses.

As is shown in FIG. 34, a trapezohedral puzzle element 42 may beoriented in two distinct directions wherein 1) one set of deltoid facesshare vertex points at the apices of the puzzle elements with the centerof the complete polyhedron and the other, apex-opposed, deltoid facesare visibly exposed or 2) the apices of the puzzle elements shared bythe previously exposed and visible faces are instead placed at thecenter of the complete polyhedron and the previously interior deltoidfaces are visibly exposed as apex-opposed faces on the exterior of thecompleted polyhedron.

Using indicia labeling for each half of each deltoid face, a flat viewof an, alternately-stellated dodecahedron with the numbered apicesshowing lower case alphabetic-labeled edges is shown in FIG. 35. Theangles of the deltoid faces have been diminished for simplicity ofpresentation in the flat view. As before, the solution and evaluationprocesses of this invention can be used in the same manner to discoversolutions for and to evaluate the difficulty of a puzzle constructedfrom trapezohedral puzzle elements.

Each trapezohedral puzzle element 42 may beneficially containtransversely-polarized magnetic attachment units on or inside eachdeltoid face as shown in FIG. 36. As with the dipyramidal puzzleelements, each deltoid face of a trapezohedral puzzle element 42 canexhibit physical characteristics corresponding with the indicium on itsco-edged deltoid face. A co-edged deltoid face is that face of thetrapezohedral puzzle element 42 with which the initial deltoid faceshares a central edge and which face is connected to the opposing apex.A central edge is an edge of the three-dimensional trapezohedral puzzleelement 42 that does not contact either apex. An example of the physicalcharacteristics of each deltoid face of a trapezohedral puzzle element42 that can be exhibited as patterns of protuberances and cavities isshown in FIG. 37. The protuberances and cavities are aligned such thatthe deltoid face of a trapezohedral puzzle element 42 interjoins withthe deltoid face of another trapezohedral puzzle element when matchingco-edged deltoid faces of the two puzzle elements are placed together attheir central edge. The patterns of protuberances and cavities preventinterjoining when an attempt is made to place two puzzle elements withnon-matching co-edged deltoid faces together. A trapezohedral puzzleelement 42 with both the transversely-polarized magnetic attachmentunits and several patterns of protuberances and cavities on the deltoidfaces is shown in FIG. 38.

Three-dimensional polyhedra with matching indicia on both sides of eachexposed central edge result when all trapezohedral puzzle elements withmatching deltoid faces are aligned and interjoined. The completedpolyhedra from trapezohedral puzzle elements are referred to in thisinvention as alternately-stellated forms.

The solution and evaluation processes of this invention may be used inthe same manner to discover solutions for and to evaluate the difficultyof an alternately-stellated dodecahedron puzzle constructed fromtrapezohedral puzzle elements in exactly the same manner as they wereused for the standard dodecahedron of the preferred embodiment.

Other alternately-stellated polyhedra may be created from3-trapezohedral, 4-trapezohedral, 6-trapezohedral, 8-trapezohedral and10-trapezohedral puzzle elements in a manner similar to those stellatedpolyhedra constructed from dipyramidal puzzle elements.

The current invention also includes geometric logic puzzlescharacterized by independent, solid, three-dimensional rhombohedricpuzzle elements with visible indicia on all identically-shapedparallelogram faces of each puzzle element that are assembled to createa three-dimensional solid polyhedron while satisfying specifiedconstraining conditions. One such puzzle constructs the rhombichexecontahedron shown in FIG. 39.

Each apex-opposed parallelogram face of each rhombohedric puzzle elementexpresses visible design indicium that can be alphanumeric values, colorelements, abstract patterns or geometric designs. Rhombohedric puzzleelements are distinguished from pyramidal, dipyramidal and trapezohedralpuzzle elements by possessing only parallelogram faces but likedipyramidal and trapezohedral puzzle elements they possess two distinctapex points and no base face. A rhombohedric puzzle element 43 is shownin FIG. 40 where the horizontal side view is shown in FIG. 40 a, a topview looking down from above one apex is shown in FIG. 40 b and a linkedvertical side view is shown in FIG. 40 c. The characteristic ratiorelating the long diagonal length i and edge length of theserhombohedric puzzle elements is specific to the completed polyhedronthat will be constructed with those puzzle elements. Each parallelogramface requires two indicia since there are two distinct edges that areshared with adjacent puzzle elements in the completed polyhedron. Anexample of the positioning of pairs of indicia on the parallelogramfaces of a rhombohedric puzzle element 43 is shown in detail in FIG. 41.The lower case alphabetic labeling of the edges of a rhombohedric puzzleelement 43 is shown in FIG. 42. Note that from above, the rhombohedricpuzzle element appears to be similar to a triangular face with a splitpair of edge labels, one on each side of each exterior edge. The anglesof the parallelogram faces have been diminished for simplicity ofpresentation. The indicia labeling of a through f shown in FIG. 42 arepresent those indicia visible when looking down on that exposed apexwith the opposed apex placed in the center of the completed polyhedron.Were the puzzle element 43 to be flipped, the indicia labeling of gthrough l would instead be visible and those indicia would be used inthe place of the exposed a through f for the solution and evaluationprocesses. Using indicia labeling for each half of each parallelogramface, a flat view of a rhombic hexecontahedron that is similar to theview of an icosahedron with the numbered faces showing lower casealphabetic-labeled split edges is shown in FIG. 43. Accordingly, thesolution and evaluation processes of this invention can be used in thesame manner to discover solutions for and to evaluate the difficulty ofa puzzle constructed from rhombohedric puzzle elements.

Similar to dipyramidal and trapezohedral puzzle elements, rhombohedricpuzzle elements may be oriented in two distinct directions wherein 1)one set of parallelogram faces share vertex points at the apices of thepuzzle elements with the center of the complete polyhedron and the otherparallelogram faces are visibly exposed or 2) the apices of the puzzleelements shared by the previously visibly exposed faces are insteadplaced at the center of the complete polyhedron and the previouslyinterior parallelogram faces are visibly exposed on the exterior of thecompleted polyhedron.

The rhombohedric puzzle elements may beneficially containtransversely-polarized magnetic, mechanical or electromechanicalattachment units on or inside each parallelogram face aligned to securetwo matching parallelogram faces when they are placed together. As withthe pyramidal and dipyramidal puzzle elements, each parallelogram faceof a rhombohedric puzzle element 43 can exhibit physical characteristicscorresponding with the indicium on its two co-edged parallelogram facesas shown in FIG. 44. Co-edged parallelogram faces are those two faces ofthe rhombohedric puzzle element 43 with which the initial parallelogramface shares a central edge and which faces are connected to the opposingapex. A central edge is an edge of the three-dimensional rhombohedricpuzzle element 43 that does not contact either apex. The physicalcharacteristics of each parallelogram face consists of patterns ofprotuberances and cavities aligned such that the face of a rhombohedricpuzzle element 43 interjoins with the parallelogram face of anotherpuzzle element when matching co-edged parallelogram faces of the twopuzzle elements are placed together along their central edges. Thepatterns of protuberances and cavities prevent interjoining when anattempt is made to place two rhombohedric puzzle elements withnon-matching co-edged parallelogram faces together. A rhombohedricpuzzle element 43 with both the transversely-polarized magneticattachment units and several patterns of protuberances and cavities onthe parallelogram faces is shown in FIG. 45.

The three-dimensional polyhedron with matching indicia on both sides ofeach exposed central edge is constructed and the puzzle completed whenall rhombohedric puzzle elements with matching parallelogram faces arealigned and interjoined. One completed polyhedra that can be createdfrom rhombohedric puzzle elements is the rhombic hexecontahedron shownin FIG. 39.

The current invention describes the use of transversely-polarizedmagnetic, mechanical or electromechanical attachment units on or insidethe attaching faces of each three-dimensional, pyramidal, dipyramidal,trapezohedral or rhombohedric puzzle element so that any puzzle elementthat is correctly aligned with an adjoining puzzle element is secured tocomplete the final polyhedra. The resulting polyhedron consists of athree-dimensional object that can be rotated, examined and handledwithout the constraint of a supporting base or structure. However, theattachment mechanism allows attaching faces of separate puzzle pieces tobe disengaged manually, as is desirable when trying to align pieces thatdo not satisfy the constraining conditions or to disassemble the puzzleto allow the challenge of reassembling it at another time. While thisspecification has specifically described the use of magnetic mechanismsfor the purpose of securing the puzzle elements, this description shouldnot be construed as a limitation on the scope of the invention.

The current invention may use physical characteristics on the attachingfaces of each pyramidal, dipyramidal, trapezohedral or rhombohedricpuzzle element consisting of patterns of protuberances and cavities,each of which cavities serve as a snugly fitting receptacle for aprotuberance, corresponding to the visual indicia of corresponding edgessuch that only properly aligned puzzle elements with matching visibleindicia on the exposed faces may be successfully interjoined. Thosepatterns of protuberances and cavities also prevent interjoining when anattempt is made to place the properly aligned faces of two non-matchingpuzzle elements together. While this specification has specificallydescribed a particular configuration for patterns of protuberances andcavities corresponding to the visual indicia of corresponding edges,this description should not be construed as a limitation on the scope ofthe invention.

Computerized processes can be created to represent puzzles of the typesdisclosed herein in visual form and allow for selection and placement ofpyramidal, dipyramidal, trapezohedral or rhombohedric puzzle elementswithin that visual representation. These computerized manifestations andrepresentations of three-dimensional polyhedra-type geometric logicpuzzles allow for the puzzles to be solved through interaction withcomputerized representations of the puzzle elements. While thisspecification specifically describes a particular approach forphysically solid, three-dimensional polyhedra-type geometric logicpuzzles, those puzzles also can be solved through interaction withcomputerized representations of the puzzle elements. Thus, thisdescription of the preferred embodiment should not be construed as alimitation on the scope of the invention.

In addition to the dodecahedron, the icosahedron, their stellatedcounterparts and the rhombic hexecontahedron described above, thecurrent invention can also be implemented to make standard and stellatedpuzzles yielding other polyhedra (shown in flat views in the notedfigures) including FIG. 46 the octahedron (from eight 3-sided,axially-symmetric, pyramidal-peaked puzzle elements), FIG. 47 thecuboctahedron (from eight 3-sided, axially-symmetric, pyramidal-peakedpyramid puzzle elements and six 4-sided, axially-symmetric,pyramidal-peaked puzzle elements), FIG. 48 the small rhombicuboctahedron(from eight 3-sided, axially-symmetric, pyramidal-peaked puzzle elementsand eighteen 4-sided, axially-symmetric, pyramidal-peaked puzzleelements), FIG. 49 the icosidodecahedron (from twenty 3-sided,axially-symmetric, pyramidal-peaked puzzle elements and twelve 5-sided,axially-symmetric, pyramidal-peaked puzzle elements), FIG. 50 therhombicosidodecahedron (from twenty 3-sided, axially-symmetric,pyramidal-peaked puzzle elements, thirty 4-sided, axially-symmetric,pyramidal-peaked puzzle elements and twelve 5-sided, axially-symmetric,pyramidal-peaked puzzle elements), FIG. 51 the truncated icosahedron(from twelve 5-sided, axially-symmetric, pyramidal-peaked puzzleelements and twenty 6-sided, axially-symmetric, pyramidal-peaked puzzleelements), FIG. 52 the great rhombicuboctahedron (from twelve 4-sided,axially-symmetric, pyramidal-peaked puzzle elements, eight 6-sided,axially-symmetric, pyramidal-peaked puzzle elements and six 8-sided,axially-symmetric, pyramidal-peaked puzzle elements) and FIG. 53 therhombitruncated icosidodecahedron (from thirty 4-sided,axially-symmetric, pyramidal-peaked puzzle elements, twenty 6-sided,axially-symmetric, pyramidal-peaked puzzle elements and twelve 10-sided,axially-symmetric, pyramidal-peaked puzzle elements).

Two other Platonic solids, the tetrahedron and the cube, could be usedas the final polyhedra for the puzzle of this invention using four3-sided, axially-symmetric, pyramidal-peaked puzzle elements and six4-sided, axially-symmetric, pyramidal-peaked puzzle elements,respectively. Since the very small number of puzzle elements makes thosetwo puzzles extremely simple, no figures or descriptions have beenincluded. However, the design of these puzzle elements and choice ofindicia for these two puzzles are essentially identical to the processesdisclosed herein for other polyhedra and the absence of detaileddescriptions should not be construed as limitations on the scope of theinvention.

This invention includes the use of visual indicia on the pyramidal,dipyramidal, trapezohedral or rhombohedric puzzle elements. In someimplementations, the visual indicia may be a compounded form of two ormore types of indicia (such as abstract patterns placed in a backgroundof color elements) with each type of indicia chosen in such a way as torepresent a distinct puzzle as is shown in FIG. 54. Five distinctshading pattern indicia are shown on a five-edged puzzle element as 44,45, 46, 47 and 48 while a combination of just three geometric designindicia are inset into the shading pattern of each edge. A squaregeometric design indicium is shown as 49 while triangular indicia areshown at 50 and 51 and octahedral indicia are shown at 52 and 53. Thesolution and evaluation processes of this invention can be to discoversolutions for and evaluate the difficulty of puzzles using combinationsof just three indicia (with some repetition of single indicia) in asimilar manner to that previously described for five distinct indicia.Solutions to such puzzles are also possible with other numbers ofdistinct visual indicia and the absence of detailed descriptions ofthese possibilities should not be construed as limitations on the scopeof the invention. A single set of such compound puzzle elements can beused to provide two or more distinctly different puzzle solvingchallenges with distinctly different levels of difficulty.

The axially-symmetric, pyramidal-peaked puzzle elements disclosed hereinincluded pyramidal, dipyramidal, trapezohedral or rhombohedric forms.Alternative three-dimensional puzzle elements for puzzles as disclosedin this invention could be created from the joining of two or more ofthe axially-symmetric, pyramidal-peaked puzzle elements disclosed hereinor from the dissection into two or more separate puzzle elements of theaxially-symmetric, pyramidal-peaked puzzle elements disclosed herein.While this specification specifically describes several examples ofaxially-symmetric, pyramidal-peaked puzzle elements, these descriptionsshould not be construed as limitations on the scope of the invention.

Although the present invention has been described in terms of thepresently preferred embodiment, it is to be understood that suchdisclosure is purely illustrative and is not to be interpreted aslimiting. Consequently, without departing from the spirit and scope ofthe invention, various alterations, modifications, and/or alternativeapplications of the invention will, no doubt, be suggested to thoseskilled in the art after having read the preceding disclosure.Accordingly, it is intended that the following claims be interpreted asencompassing all alternations, modifications, or alternativeapplications as fall within the true spirit and scope of the invention.

1. A puzzle, comprising: a. a plurality of independent,three-dimensional, axially-symmetric, pyramidal puzzle elements, b. eachpuzzle element having a regular polygonal base with a base face definedby a plurality of symmetric base edges and radial faces extending fromeach base edge to a pyramidal peak, c. each base edge being marked withdesired indicia, d. at least one matching condition for matching indiciaon adjoining base edges of adjoining puzzle elements, e. whereinsuccessful completion of the puzzle is achieved when said puzzleelements are placed in an adjoining manner so that each base edge ofeach puzzle element adjoins an adjoining base edge of an adjoiningpuzzle piece to construct a complete polyhedron and the indicia of eachbase edge of each puzzle element satisfies the matching condition withrespect to the indicia on the adjoining base edge of the adjoiningpuzzle element.
 2. A puzzle according to claim 1, further comprising: f.attachment means for releaseably attaching each puzzle element to eachadjoining puzzle element.
 3. A puzzle according to claim 2, wherein saidattachment means comprises transversely-polarized magnets attached toeach radial face of each puzzle element so as to releaseably attach saidradial face of said puzzle element to a correctly aligned radial face ofeach other puzzle element.
 4. A puzzle according to claim 2, whereinsaid attachment means comprises transversely-polarized magnets attachedunder each radial face of each puzzle element so as to releaseablyattach said radial face of said puzzle element to a correctly alignedradial face of each other puzzle element.
 5. A puzzle according to claim2, wherein said attachment means comprises a mechanical attachmentmechanism.
 6. A puzzle according to claim 2, wherein said attachmentmeans comprises an electro-mechanical attachment mechanism.
 7. A puzzleaccording to claim 1, wherein said indicia comprises an element from aset of selected visual patterns, wherein each element of said setappears on at least one base edge.
 8. A puzzle according to claim 7,wherein each visual pattern comprises a particular color.
 9. A puzzleaccording to claim 7, wherein each visual pattern comprises a particularalphanumeric character.
 10. A puzzle according to claim 7, wherein eachvisual pattern appears on a base edge of at least two puzzle elementsand said matching condition is to adjoin base edges marked with samevisual patterns.
 11. A puzzle according to claim 7, wherein each baseedge is marked with a first element from a first set of selected visualpatterns and a second element from a second set of selected visualpatterns, and successful completion of the puzzle is achieved when saidpuzzle elements are placed in an adjoining manner so that each base edgeof each puzzle element adjoins an adjoining base edge of an adjoiningpuzzle piece to construct a complete polyhedron and either: i. the firstelement marked on each base edge of each puzzle element satisfies afirst matching condition with respect to the first element marked on theadjoining base edge of the adjoining puzzle element or ii. the secondelement marked of each base edge of each puzzle element satisfies asecond matching condition with respect to the second element marked onthe adjoining base edge of the adjoining puzzle element.
 12. A puzzleaccording to claim 1, wherein a radial face of a puzzle element has atleast one physical protuberance and one receptacle for a protuberancecorresponding to a protuberance and a receptacle for a protuberance onan adjoining radial face of an adjoining puzzle element.
 13. A puzzleaccording to claim 12, wherein said physical protuberance of said radialface of said puzzle element fits into said receptacle for a protuberanceon an adjoining radial face of an adjoining puzzle element when saidpuzzle element and said adjoining puzzle element adjoin only when thematching condition is satisfied.
 14. A puzzle, comprising: a. aplurality of independent, three-dimensional, axially-symmetric,dipyramidal puzzle elements, b. each puzzle element having a centralbisecting plane of regular polygonal shape defined by a plurality ofsymmetric central edges and radial faces extending from each centraledge in opposite directions to two pyramidal peaks, c. each radial facebeing marked with desired indicia, d. at least one matching conditionfor matching indicia on adjoining radial faces of adjoining puzzleelements, e. wherein successful completion of the puzzle is achievedwhen said puzzle elements are placed in an adjoining manner so that eachradial face of each puzzle element adjoins an adjoining radial face ofan adjoining puzzle piece to construct a stellated polyhedron and theindicia of each radial face of each puzzle element satisfies thematching condition with respect to the indicia on the adjoining radialface of the adjoining puzzle element.
 15. A puzzle according to claim14, further comprising: f. attachment means for releaseably attachingeach puzzle element to each adjoining puzzle element.
 16. A puzzleaccording to claim 15, wherein said attachment means comprisestransversely-polarized magnets attached to each radial face of eachpuzzle element so as to releaseably attach said radial face of saidpuzzle element to a correctly aligned radial face of each other puzzleelement.
 17. A puzzle according to claim 15, wherein said attachmentmeans comprises transversely-polarized magnets attached under eachradial face of each puzzle element so as to releaseably attach saidradial face of said puzzle element to a correctly aligned radial face ofeach other puzzle element.
 18. A puzzle according to claim 14, whereinsaid indicia comprises an element from a set of selected visualpatterns, wherein each element of said set appears on at least oneradial face.
 19. A puzzle according to claim 18, wherein each visualpattern appears on a radial face of at least two puzzle elements andsaid matching condition is to adjoin radial faces marked with samevisual patterns.
 20. A puzzle according to claim 14, wherein a radialface of a puzzle element has at least one physical protuberance and onereceptacle for a protuberance corresponding to a protuberance and areceptacle for a protuberance on an adjoining radial face of anadjoining puzzle element.
 21. A puzzle according to claim 20, whereinsaid physical protuberance of said radial face of said puzzle elementfits into said receptacle for a protuberance on an adjoining radial faceof an adjoining puzzle element when said puzzle element and saidadjoining puzzle element adjoin only when the matching condition issatisfied.
 22. A puzzle, comprising: a. a plurality of independent,three-dimensional, axially-symmetric, trapezohedral puzzle elements, b.each puzzle element having a central bisecting plane of regularpolygonal shape and deltoid faces each with two central edges, whichdeltoid faces extend from each central edge in opposite directions totwo pyramidal peaks, c. each central edge being marked with desiredindicia, d. at least one matching condition for matching indicia onadjoining central edges of adjoining puzzle elements, e. whereinsuccessful completion of the puzzle is achieved when said puzzleelements are placed in an adjoining manner so that each central edge ofeach puzzle element adjoins an adjoining central edge of an adjoiningpuzzle piece to construct an alternately-stellated polyhedron and theindicia of each central edge of each puzzle element satisfies thematching condition with respect to the indicia on the adjoining centraledge of the adjoining puzzle element.
 23. A puzzle, comprising: a. aplurality of independent, three-dimensional, axially-symmetric,rhombohedric puzzle elements, b. each puzzle element having a centralbisecting plane of regular polygonal shape and parallelogram faces eachwith two central edges, which parallelogram faces extend from eachcentral edge in opposite directions to two pyramidal peaks, c. eachcentral edge being marked with desired indicia, d. at least one matchingcondition for matching indicia on adjoining central edges of adjoiningpuzzle elements, e. wherein successful completion of the puzzle isachieved when said puzzle elements are placed in an adjoining manner sothat each central edge of each puzzle element adjoins an adjoiningcentral edge of an adjoining puzzle piece to construct a stellatedpolyhedron and the indicia of each central edge of each puzzle elementsatisfies the matching condition with respect to the indicia on theadjoining central edge of the adjoining puzzle element.
 24. A processfor evaluating indicia to be exhibited on three-dimensional puzzleelements to enable selection of indicia that will allow puzzle elementsto be assembled in a manner that satisfies a matching condition for suchindicia, comprising the steps of: a. assign a separate identifyingmoniker to each face of each puzzle element, b. select an initial puzzlepiece, c. list, in adjacent order, the monikers assigned to each face ofsaid initial puzzle piece, d. select a second puzzle element, e.determine whether said second puzzle element can be placed adjacent tosaid initial puzzle piece in a manner which satisfies said matchingcondition, f. list, in adjacent order, the monikers assigned to eachface of said second puzzle piece in any manner which, when adjacent tosaid initial puzzle piece, satisfies said matching condition, g. repeatsteps d-f for each remaining puzzle piece, h. list and count allpossible arrangements of puzzle elements in a manner that satisfies saidmatching condition, j. analyze whether any of the possible arrangementsof puzzle elements satisfies said matching condition.
 25. A process forevaluating indicia to be exhibited on three-dimensional puzzle elementsto enable selection of indicia that will allow puzzle elements to beassembled in a manner that satisfies a matching condition for suchindicia with a desired degree of difficulty, comprising the steps of: a.assign a separate identifying moniker to each face of each puzzleelement, b. select an initial puzzle piece, c. list, in adjacent order,the monikers assigned to each face of said initial puzzle piece, d.select a second puzzle element, e. determine whether said second puzzleelement can be placed adjacent to said initial puzzle piece in a mannerwhich satisfies said matching condition, f. list, in adjacent order, themonikers assigned to each face of said second puzzle piece in any mannerwhich, when adjacent to said initial puzzle piece, satisfies saidmatching condition, g. repeat steps d-f for each remaining puzzle piece,h. list and count all possible arrangements of puzzle elements, i. listand count all possible arrangements of puzzle elements in a manner thatsatisfies said matching condition, j. analyze whether the ratio ofpossible arrangements of puzzle elements in a manner that satisfies saidmatching condition to the possible arrangement of puzzle elementsindicates said desired degree of difficulty.
 26. A computerized processrepresenting three dimensional puzzle elements in visual form allowingfor selection and placement of the puzzle elements within a visualrepresentation for completion of a specified puzzle.